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## Science Modules / Pathogens

### Overview

Understanding the fate and transport of pathogenic and indicator microbes within drinking water reservoirs is critical for managers to effectively reduce risk. Over the past decade several advances have been made for the simulating organism dynamics using coupled hydrodynamic-organism fate models. These have generally been used for simulating coliform bacteria (Hipsey et al., 2008), with applications also reported for Cryptosporidium (Hipsey et al., 2004) and viruses (Sokolova et al., 2012). In general, these models simulate organism concentrations within water bodies by accounting for external loading, advection and mixing process that occur within the lake interior, organism inactivation and sedimentation. A synthesis of relevant information from empirical and prior modelling studies is presented on the adjacent tab to define the model approach for river/reservoir or coastal systems, and summarises the implementation of the pathogen module within the AED2 model framework.

Organisms that can be simulated using the AED2 framework include:

• Protozoa
• • Cryptosporidium
• Giardia
• Bacteria
• • E coli
• Enterococci
• Campylobacter (C. jejuni species)
• Viruses
• • Norovirus
• Rotavirus

A table of parameters for each of the above groups and other common organism classes is provided below with indicative references, and additionally a example database of parameters configured in the appropriate input file format are available via the button below.

### General Approach

The general balance equation for organism transport and fate is summarized in Hipsey et al., (2008) as:

Eq1

where C is the organic concentration (orgs m-3), t is time, xj is the distance in the j-th dimension (m), Uj is the velocity in the j-th dimension (m s-1), is the eddy-diffusivity and Cin and Cout are the inflow and outflow fluxes respectively (orgs m-3 s-1). This relates organism concentration through time to hydrodynamic characteristics within the lake (as captured by U, and ), and the environmental conditions experienced that influence organism fate (including temperature, salinity, light intensity, dissolved organic carbon, oxygen and pH). This comprehensive description is usually simplified for specific applications, based on justification of the dominant processes present in any particular lake and available data for model setup and validation. Within the TUFLOWFV-AED2 model framework, the 1st three terms on the RHS are solved via the finite volume scalar transport routines with TUFLOW-FV and the fourth and fifth terms are simulated by the aed2_pathogens module of the AED2 aquatic ecological modelling library. For this application the growth and predation terms suggested by Hipsey et al. (2008) are not considered directly since the likelihood of growth is small (Toze et al., 2012) and predation/grazing is factored into the die-off rate (described below). The remaining terms for simulation of organism resuspension and inactivation are described in detail next.

### Natural Mortality

Natural mortality, or the ‘dark-death rate’, kd, is an important process determining the net rate of die off of protozoan, viral and bacterial organisms and has been reported to vary for specific organisms due to changes in temperature, salinity and pH. The reported die off rates in the literature however are widely variable, with a synthesis of numerous studies from a range of water bodies presented in Hipsey et al., (2008). For freshwater reservoirs, changes in salinity and pH are unlikely to be a significant driver of organism viability relative to the range presented Hipsey et al., (2008) and therefore a simple constant die-off rate that depends on temperature is appropriate:

Eq2

where is the rate of mortality in the dark at 20C and in freshwater. Since the AED2 implementation of the model to be applied with NPD does not include protozoan grazing as a separate term (as in Eq 1), the grazing effect needs to be embodied within the term. This effectively assumes a constant grazing pressure over time, and if chosen to be a low value this will essentially ensure conservative estimate of the die-off due to grazing. Empirical data from Wivenhoe dam shows the presence of native micro-organisms can increase the background die-off rate by 1.1-3.0x (eg Table 5 in Toze et al., 2012).

### Sunlight inactivation

Depending on the clarity of the water, the light climate of the lake can be a dominant factor influencing organism viability and this has been observed empirically in Wivenhoe Dam by Toze et al. (2012). Different organism types experience different sensitivity to different light bandwidths, with most organisms sensitive to UV-B and UV-A and some sensitive to light in the visible spectrum (Sinton eta l). Hipsey et al. (2008) formulated a multiple band-width model for organisms that included direct and indirect mechanisms for sunlight mediated inactivation by accounting for the effect of salinity, oxygen and pH on free radical formation. Here we use a simplified form that accounts only for direct sunlight denaturation as the indirect mechanism is more specific to MS2 phage relative to rotavirus for example (Verbyla et al., 2015). The implemented expression is therefore:

Eq3

where NB is the number of discrete solar bandwidths to be modeled, b is the bandwidth class {1, 2, … , NB}, kb is the freshwater inactivation rate coefficient for exposure to the bth class (m2 MJ-1), Ib is the intensity of the bth bandwidth class (Wm-2),  is a constant to convert units from seconds to days and J to MJ (= 8.6410-2). In AED2, the light intensity is computed for 3 distinct bandwidths, including UV-B, UV-A and PAR, and the attenuation of each through the reservoir water column is based on bandwidth specific light extinction coefficients, that account for the effect of turbidity and chromophoric dissolved organic matter (CDOM) on attenuation.

### Sedimentation

Sedimentation of organisms occurs at a rate depending on the degree to which the population is attached to suspended particles. Within AED2, this is captured by simulating free and attached organisms and multiple groups of particles may be accounted for. If we assume a single dominant particle size and ignore the effect of salinity on the settling velocity, then the expression in Hipsey et al (2008) for the effective sedimentation velocity reduces to:

Eq4

where fa is the attached fraction, and V is the vertical velocity of organisms or sediment particles.

### Resuspension

Resuspension of organisms accumulated within the sediment has been show to be a relatively important terms in environments where high currents or waves exist. In reservoirs this can occur in the lake margins or deeper in the lake where internal waves or river underflows increase the shear stress at the sediment. The amount of organisms that resuspend depends not only on the shear stress but also n the concentration of organisms in the surficial layers of the sediment. This may be modelled by accounting for the deposited organisms, decay within the sediment and resuspension rate, however, this is notoriously difficult given potentially complex dynamics of organisms in the sediment. Instead we assume that a standard background concentration exists within different sediment regions (based on depth and local geomorphology) and simulate resuspension rate as:

Eq5

where, is the rate of organism suspension (orgs m-2 s-1), which occurs when the critical shear stress is exceeded in the relative computational cell.

#### Variable Summary & Setup Options

Variable Name Description Units Variable Type Core/Optional
PTH_name Concentration of pathogens cfu/100mL pelagic optional
to be completed
Variable Name Description Units Variable Type Core/Optional
Parameter Name Description Units Parameter Type Default Typical Range Comment
num_pathogens number of pathogens to model 1-?
to be completed

An example nml block for the pathogens module is shown below.

 &aed2_pathogens num_pathogens = 2 the_pathogens = 1,3 dbase = './the_path_to/aed2_pathogen_pars.nml' ! OPTIONAL VARS HERE resuspension num_ss ss_set ss_tau ss_ke sim_sedorgs oxy_variable epsilon tau_0 tau_0_min Ktau_0 extra_diag att_ts /

2018 : aed2_pathogen_pars.nml parameter formatting style

 !------------------------------------------------------------------------------- ! aed2_pathogen_pars.nml : PATHOGEN PARAMETER DATABASE !------------------------------------------------------------------------------- ! p_name : [ string] - pathogen name ! coef_grwth_uMAX : [ real] - Max growth rate at 20C ! coef_grwth_Tmin : [ real] - Tmin and Tmax, f(T) ! coef_grwth_Tmax : [ real] - Tmin and Tmax, f(T) ! coef_grwth_T1 : [ real] - coef_grwth_T1 and coef_grwth_T2 ! coef_grwth_T2 : [ real] - coef_grwth_T1 and coef_grwth_T2 ! coef_grwth_Kdoc : [ real] - Half-saturation for growth, coef_grwth_Kdoc ! coef_grwth_ic : [ real] - coef_grwth_ic ! coef_mort_kd20 : [ real] - Mortality rate (Dark death rate) @ 20C and 0 psu ! coef_mort_theta : [ real] - Temperature multiplier for mortality: coef_mort_theta ! coef_mort_c_SM : [ real] - Salinity effect on mortality ! coef_mort_alpha : [ real] - Salinity effect on mortality ! coef_mort_beta : [ real] - Salinity effect on mortality ! coef_mort_c_PHM : [ real] - pH effect on mortality ! coef_mort_K_PHM : [ real] - pH effect on mortality ! coef_mort_delta_M : [ real] - pH effect on mortality ! coef_mort_fdoc : [ real] - Fraction of mortality back to doc ! coef_light_kb_vis : [ real] - Light inactivation ! coef_light_kb_uva : [ real] - Light inactivation ! coef_light_kb_uvb : [ real] - Light inactivation ! coef_light_cSb_vis : [ real] - Salinity effect on light inactivation ! coef_light_cSb_uva : [ real] - Salinity effect on light inactivation ! coef_light_cSb_uvb : [ real] - Salinity effect on light inactivation ! coef_light_kDOb_vis : [ real] - DO effect on light ! coef_light_kDOb_uva : [ real] - DO effect on light ! coef_light_kDOb_uvb : [ real] - DO effect on light ! coef_light_cpHb_vis : [ real] - pH effect on light inactivation ! coef_light_cpHb_uva : [ real] - pH effect on light inactivation ! coef_light_cpHb_uvb : [ real] - pH effect on light inactivation ! coef_light_KpHb_vis : [ real] - pH effect on light inactivation ! coef_light_KpHb_uva : [ real] - pH effect on light inactivation ! coef_light_KpHb_uvb : [ real] - pH effect on light inactivation ! coef_light_delb_vis : [ real] - exponent for pH effect on light inactivation ! coef_light_delb_uva : [ real] - exponent for pH effect on light inactivation ! coef_light_delb_uvb : [ real] - exponent for pH effect on light inactivation ! coef_pred_kp20 : [ real] - Loss rate due to predation and temp multiplier ! coef_pred_theta_P : [ real] - Loss rate due to predation and temp multiplier ! coef_sett_fa : [ real] - Attached fraction in water column ! coef_sett_w_path : [ real] - Sedimentation velocity (m/d) at 20C (-ve means down) for NON-ATTACHED orgs ! coef_resus_epsilonP : [ real] - Pathogen resuspension rate ! coef_resus_tauP_0 : [ real] - Critical shear stress for organism resuspension &pathogen_data pd%p_name = 'crypto', 'ecoli', 'totalcoli' pd%coef_grwth_uMAX = 0, 3, 2.4 pd%coef_grwth_Tmin = 4, 4, 4 pd%coef_grwth_Tmax = 35, 35, 35 pd%coef_grwth_T1 = 0.008, 0.008, 0.008 pd%coef_grwth_T2 = 0.1, 0.1, 0.1 pd%coef_grwth_Kdoc = 0, 0.3, 0.3 pd%coef_grwth_ic = 1.0E-9, 1.0E-9, 1.0E-9 pd%coef_mort_kd20 = 0.03, 0.48, 0.34 pd%coef_mort_theta = 1.08, 1.08, 1.11 pd%coef_mort_c_SM = 0, 2.0E-10, 2.0E-7 pd%coef_mort_alpha = 0, 6.1, 4.2 pd%coef_mort_beta = 0, 0.25, 0.25 pd%coef_mort_c_PHM = 0, 50, 50 pd%coef_mort_K_PHM = 0, 6, 6 pd%coef_mort_delta_M = 0, 4, 4 pd%coef_mort_fdoc = 0, 0.5, 0.5 pd%coef_light_kb_vis = 0, 0.097, 0.097 pd%coef_light_kb_uva = 0, 1.16, 1.16 pd%coef_light_kb_uvb = 0, 36.4, 36.4 pd%coef_light_cSb_vis = 0.0067, 0.0067, 0.0067 pd%coef_light_cSb_uva = 0.0067, 0.0067, 0.0067 pd%coef_light_cSb_uvb = 0.0067, 0.0067, 0.0067 pd%coef_light_kDOb_vis = 0.5, 0.5, 0.5 pd%coef_light_kDOb_uva = 0.5, 0.5, 0.5 pd%coef_light_kDOb_uvb = 0.5, 0.5, 0.5 pd%coef_light_cpHb_vis = 10, 10, 10 pd%coef_light_cpHb_uva = 10, 10, 10 pd%coef_light_cpHb_uvb = 10, 10, 10 pd%coef_light_KpHb_vis = 5, 5, 5 pd%coef_light_KpHb_uva = 5, 5, 5 pd%coef_light_KpHb_uvb = 5, 5, 5 pd%coef_light_delb_vis = 3, 3, 3 pd%coef_light_delb_uva = 3, 3, 3 pd%coef_light_delb_uvb = 3, 3, 3 pd%coef_pred_kp20 = 0, 0.2, 0.2 pd%coef_pred_theta_P = 1, 1.04, 1.04 pd%coef_sett_fa = 0, 0.94, 0.81 pd%coef_sett_w_path = -2.5E-6, -5.0E-7, -5.0E-7 pd%coef_resus_epsilonP = 0.01, 0.01, 0.01 pd%coef_resus_tauP_0 = 0.01, 0.01, 0.01 /

2014 : aed2_pathogen_pars.nml style parameters (note: not compatible with the online parameter database)

 !----------------------------------------------------------------! ! coef_grwth_uMAX !-- Max growth rate at 20C ! coef_grwth_Tmin, coef_grwth_Tmax !-- Tmin and Tmax, f(T) ! coef_grwth_T1, coef_grwth_T2 !-- coef_grwth_T1 and coef_grwth_T2 ! coef_grwth_Kdoc !-- Half-saturation for growth, coef_grwth_Kdoc ! coef_grwth_ic !-- coef_grwth_ic ! coef_mort_kd20 !-- Mortality rate (Dark death rate) @ 20C and 0 psu ! coef_mort_theta !-- Temperature multiplier for mortality: coef_mort_theta ! coef_mort_c_SM, coef_mort_alpha, coef_mort_beta !-- Salinity effect on mortality ! coef_mort_c_PHM, coef_mort_K_PHM, coef_mort_delta_M !-- pH effect on mortality ! coef_mort_fdoc !-- Fraction of mortality back to doc ! coef_light_kb_vis, coef_light_kb_uva, coef_light_kb_uvb !-- Light inactivation ! coef_light_cSb_vis, coef_light_cSb_uva, coef_light_cSb_uvb !-- Salinity effect on light inactivation ! coef_light_kDOb_vis, coef_light_kDOb_uva, coef_light_kDOb_uvb !-- DO effect on light ! coef_light_cpHb_vis, coef_light_cpHb_uva, coef_light_cpHb_uvb !-- pH effect on light inactivation ! coef_light_KpHb_vis, coef_light_KpHb_uva, coef_light_KpHb_uvb !-- pH effect on light inactivation ! coef_light_delb_vis, coef_light_delb_uva, coef_light_delb_uvb !-- exponent for pH effect on light inactivation ! coef_pred_kp20, coef_pred_theta_P !-- Loss rate due to predation and temp multiplier ! coef_sett_fa !-- Attached fraction in water column ! coef_sett_w_path !-- Sedimentation velocity (m/d) at 20C (-ve means down) for NON-ATTACHED orgs !----------------------------------------------------------------! ! p_name coef_grwth_uMAX, coef_grwth_Tmin, coef_grwth_Tmax, coef_grwth_T1, coef_grwth_T2, coef_grwth_Kdoc, coef_grwth_ic, coef_mort_kd20, coef_mort_theta, coef_mort_c_SM, coef_mort_alpha, coef_mort_beta, coef_mort_c_PHM, coef_mort_K_PHM, coef_mort_delta_M, coef_mort_fdoc, coef_light_kb_vis, coef_light_kb_uva, coef_light_kb_uvb, coef_light_cSb_vis, coef_light_cSb_uva, coef_light_cSb_uvb, coef_light_kDOb_vis, coef_light_kDOb_uva, coef_light_kDOb_uvb, coef_light_cpHb_vis, coef_light_cpHb_uva, coef_light_cpHb_uvb, coef_light_KpHb_vis, coef_light_KpHb_uva, coef_light_KpHb_uvb, coef_light_delb_vis, coef_light_delb_uva, coef_light_delb_uvb, coef_pred_kp20, coef_pred_theta_P, coef_sett_fa, coef_sett_w_path &pathogen_data pd = 'crypto', 0.0, 4.0, 35.0, 0.008, 0.1, 0.0, 1e-9, 0.03, 1.14, 0.00, 0.0, 0.00, 0.0, 0.0, 0.0, 0.0, 0.000, 2.13, 33.7, 0.0067, 0.0067, 0.0067, 0.5, 0.5, 0.5, 10.0, 10.0, 10.0, 5.0, 5.0, 5.0, 3.0, 3.0, 3.0, 0.0, 1.00, 0.00, -2.\5e-6, 'ecoli', 0.0, 4.0, 35.0, 0.008, 0.1, 0.3, 1e-9, 0.48, 1.08, 2e-10, 6.1, 0.25, 50.0, 6.0, 4.0, 0.5, 0.097, 1.16, 36.4, 0.0067, 0.0067, 0.0067, 0.5, 0.5, 0.5, 10.0, 10.0, 10.0, 5.0, 5.0, 5.0, 3.0, 3.0, 3.0, 0.0, 1.04, 0.94, -0.5e-6, 'fcoli', 0.0, 4.0, 35.0, 0.008, 0.1, 0.3, 1e-9, 0.71, 1.06, 2e-3, 1.8, 0.25, 50.0, 6.0, 4.0, 0.5, 0.097, 1.16, 36.4, 0.0067, 0.0067, 0.0067, 0.5, 0.5, 0.5, 10.0, 10.0, 10.0, 5.0, 5.0, 5.0, 3.0, 3.0, 3.0, 0.0, 1.04, 0.81, -0.5e-6, 'ent', 0.0, 4.0, 35.0, 0.008, 0.1, 0.3, 1e-9, 0.45, 1.04, 0.00, 0.0, 0.25, 50.0, 6.0, 4.0, 0.5, 0.882, 1.16, 17.2, 0.0067, 0.0067, 0.0067, 0.5, 0.5, 0.5, 10.0, 10.0, 10.0, 5.0, 5.0, 5.0, 3.0, 3.0, 3.0, 0.0, 1.04, 0.81, -0.5e-6, 'totalcoli', 0.0, 4.0, 35.0, 0.008, 0.1, 0.3, 1e-9, 0.34, 1.11, 2e-7, 4.2, 0.25, 50.0, 6.0, 4.0, 0.5, 0.097, 1.16, 36.4, 0.0067, 0.0067, 0.0067, 0.5, 0.5, 0.5, 10.0, 10.0, 10.0, 5.0, 5.0, 5.0, 3.0, 3.0, 3.0, 0.0, 1.04, 0.81, -0.5e-6, /

pending ......

#### Publications & References

Auer, M. T., and S. L. Niehaus (1993), Modeling fecal coliform bacteria – I. Field and laboratory determination of loss kinetics, Water Res., 27, 693-701.

Bertrand, I. et al. (2012), The impact of temperature on the inactivation of enteric viruses in food and water: a review, Journal of Applied Microbiology, 112(6), 1059–1074.

Bae, J., and K. J. Schwab (2008), Evaluation of Murine Norovirus, Feline Calicivirus, Poliovirus, and MS2 as Surrogates for Human Norovirus in a Model of Viral Persistence in Surface Water and Groundwater, Applied and Environmental Microbiology, 74(2), 477–484.

Cervantes, A. A. (2012), Resuspension of E. coli Under Controlled Flows and Stream Bottom Sediments, MSc Thesis , Iowa State University. 167 pp

Silva, A. K. D., F. S. L. Guyader, J.-C. L. Saux, M. Pommepuy, M. A. Montgomery, and M. Elimelech (2008), Norovirus Removal and Particle Association in a Waste Stabilization Pond, Environ. Sci. Technol., 42(24), 9151–9157.

Dai, X., and J. Boll (2006), Settling velocity of Crytposporidium parvum and Giardia lamblia, Water Res., 40, 1321-1325.

Gerba, C. P., and G. E. Schaiberger (1975), Effect of particulates on virus survival in seawater, J. Water Pollut. Control Fed., 47, 93-103.

Ge, Z., R. L. Whitman, M. B. Nevers, M. S. Phanikumar, and M. N. Byappanahalli (2012), Nearshore hydrodynamics as loading and forcing factors for Escherichia coli contamination at an embayed beach, Limnol. Oceanogr., 57(1), 362–381, doi:10.4319/lo.2012.57.1.0362.

Hipsey, M. R., J. P. Antenucci, J. D. Brookes, M. D. Burch, R. H. Regel, and L. Linden (2004), A three dimensional model of Cryptosporidium dynamics in lakes and reservoirs: a new tool for risk management, Intl. J. River Basin Management, 2, 181-197.

Hipsey, M. R., J. D. Brookes, R. H. Regel, J. P. Antenucci, and M. D. Burch (2006), In situ evidence for the association of total coliforms and Escherichia coli with suspended inorganic particles in an Australian reservoir, Water Air Soil Pollut., 170, 191-209.

Hipsey, M. R., J. P. Antenucci, and J. D. Brookes (2008), A generic, process-based model of microbial pollution in aquatic systems, Water Resour. Res., 44, W07408.

King, B. J., D. Hoefel, D. P. Daminato, S. Fanok, and P. T. Monis (2007), Solar UV dramatically reduces Cryptosporidium parvum oocyst infectivity in environmental waters, Appl. Environ. Microbiol., in review.

King, B. J., A. R. Keegan, P. T. Monis, and C. P. Saint (2005), Environmental temperature controls Cryptosporidium oocyst metabolic rate and associated retention of infectivity, Appl. Environ. Microbiol., 71, 3848-3857.

Meng, Z.-D., C. Burch, R. Heath, and I. Gust (1987), Physicochemical stability and inactivation of human and simian rotaviruses, Applied and Environmental Microbiology, 53(4), 727–730.

Medema, G. J., F. M. Schets, P. F. M. Teunis, and A. H. Havelaar (1998), Sedimentation of free and attached Cryptosporidium oocysts and Giardia cysts in water, Appl. Environ. Microbiol., 64, 4460-4466.

Pandey, P. K., M. L. Soupir, and C. R. Rehmann (2012), A model for predicting resuspension of Escherichia coli from streambed sediments, Water Res., 46(1), 115–126.

Payment, P., E. Morin, and M. Trudel (1988), Coliphages and enteric viruses in the particulate phase of river water, Can. J. Microbiol., 34, 907-910.

Rodríguez, S., and R. Araujo (2012), Effect of environmental parameters on the inactivation of the waterborne pathogen Campylobacter in a Mediterranean river, J Water Health, 10(1), 100–10.

Romero, O. C., A. P. Straub, T. Kohn, and T. H. Nguyen (2011), Role of Temperature and Suwannee River Natural Organic Matter on Inactivation Kinetics of Rotavirus and Bacteriophage MS2 by Solar Irradiation, Environ. Sci. Technol., 45(24), 10385–10393

Searcy, K. E., A. I. Packman, E. R. Atwill, and T. Harter (2005), Association of Cryptosporidium parvum with suspended particles: Impact on oocyst sedimentation, Applied and Environmental Microbiology, 71(2), 1072–1078.

Sidhu, J. P. S., and S. Toze (2012), Assessment of pathogen survival potential during managed aquifer recharge with diffusion chambers, J Appl Microbiol, 113(3), 693–700.

Sinton, L. W., C. H. Hall, P. A. Lynch, and R. J. Davies-Colley (2002), Sunlight inactivation of fecal indicator bacteria and bacteriophages from waste stabilization pond effluent in fresh and saline waters, Appl. Environ. Microbiol., 68, 1122-1131.

Sinton, L., C. Hall, and R. Braithwaite (2007), Sunlight inactivation of Campylobacter jejuni and Salmonella enterica, compared with Escherichia coli, in seawater and river water, Journal of Water and Health, 5(3), 357–9,.

Sokolova, E., J. Åström, T. J. R. Pettersson, O. Bergstedt, and M. Hermansson (2012), Estimation of pathogen concentrations in a drinking water source using hydrodynamic modelling and microbial source tracking, J Water Health, 10(3), 358–13,.

Toze, S., Hodgers, L., Palmer, A., Sidhu, J., Page, D., Williams, M., Kookana, R., Bartkow, M., Sedlak, D., Stratton, H., Ahmed, W., Schroeder, S. and Christie, M. (2012). Natural Attenuation of Pathogens and Trace Contaminants in South East Queensland Waterways. Urban Water Security Research Alliance Technical Report No. 87.

Verbyla, M. E., and J. R. Mihelcic (2015), A review of virus removal in wastewater treatment pond systems, Water Res., 71(C), 107–124.

Walker, F. R., and J. R. Stedinger (1999), Fate and transport model of Cryptosporidium, J. Environ. Eng., 125, 325-333.

Yang, Y., and M. W. Griffiths (2013), Comparative Persistence of Subgroups of F-Specific RNA Phages in River Water, Applied and Environmental Microbiology, 79(15), 4564–4567


text:
# Derived Functors of a Half-Exact Functor

This question is mainly posed out of sheer curiosity rather than any practical applications.

An additive functor F:C→DF:C\to D is called half-exact if for every short exact sequence
0→L→M→N→0
0\to L\to M\to N\to 0

There is an exact sequence
FL→FM→FN
FL\to FM\to FN

When FF is in fact left exact (resp. right exact), one can then define the right derived functors (resp. left derived functors) RiFR^i F (resp. LiFL_i F). These create long exact sequences which “fill in” the missing exactness in a “natural” way (i.e., in such a way that morphisms of short exact sequences induce morphisms of long exact sequences).

I am interested in the generalization to the case where FF is neither left exact nor right exact. Namely: do there exist derived functors RiFR^i F and LiFL_i F such that i) RiF=L−iFR^i F = L_{-i} F, and ii) for every short exact sequence of objects in CC there is an induced long exact sequence
…→L1FN→L→M→N→R1FL→…
…\to L_1 FN\to L\to M\to N\to R_1 FL\to …

which is natural in the sense mentioned above? And if so, how would one go about defining these functors?

=================

related question math.stackexchange.com/questions/1507311/half-exact-functor . Instead of “there is an exact sequence…” you should say “the sequence… is exact”. Obviously you want that the maps in second sequence are induced from those of the first one?
– syzygy
2 days ago

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text:
# 9.2 Outcomes and the type i and type ii errors

 Page 1 / 12

When you perform a hypothesis test, there are four possible outcomes depending on the actual truth (or falseness) of the null hypothesis H 0 and the decision to reject or not. The outcomes are summarized in the following table:

STATISTICAL DECISION H 0 IS ACTUALLY...
True False
Cannot reject H 0 Correct Outcome Type II error
Cannot accept H 0 Type I Error Correct Outcome

The four possible outcomes in the table are:

1. The decision is cannot reject H 0 when H 0 is true (correct decision).
2. The decision is cannot accept H 0 when H 0 is true (incorrect decision known as a Type I error ). This case is described as "rejecting a good null". As we will see later, it is this type of error that we will guard against by setting the probability of making such an error. The goal is to NOT take an action that is an error.
3. The decision is cannot reject H 0 when, in fact, H 0 is false (incorrect decision known as a Type II error ). This is called "accepting a false null". In this situation you have allowed the status quo to remain in force when it should be overturned. As we will see, the null hypothesis has the advantage in competition with the alternative.
4. The decision is cannot accept H 0 when H 0 is false ( correct decision whose probability is called the Power of the Test ).

Each of the errors occurs with a particular probability. The Greek letters α and β represent the probabilities.

α = probability of a Type I error = P (Type I error) = probability of rejecting the null hypothesis when the null hypothesis is true.

β = probability of a Type II error = P (Type II error) = probability of not rejecting the null hypothesis when the null hypothesis is false.

α and β should be as small as possible because they are probabilities of errors.

Statistics allows us to set the probability that we are making a Type I error. The probability of making a Type I error is α. Recall that the confidence intervals in the last unit were set by choosing a value called Z α (or t α ) and the alpha value determined the confidence level of the estimate because it was the probability of the interval capturing the true mean (or proportion parameter p). This alpha and that one are the same.

The easiest way to see the relationship between the alpha error and the level of confidence is with the following figure.

In the center of [link] is a normally distributed probability distribution marked Ho. This is a sampling distribution of $\stackrel{–}{X}$ and by the Central Limit Theorem it is normally distributed. The distribution in the center is marked H 0 and represents the distribution for the null hypotheses H 0 : µ = 100. This is the value that is being tested. The formal statements of the null and alternative hypotheses are listed below the figure.

The distributions on either side of the H 0 distribution represent distributions that would be true if H 0 is false, under the alternative hypothesis listed as H a . We do not know which is true, and will never know. There are, in fact, an infinite number of distributions from which the data could have been drawn if H a is true, but only two of them are on [link] representing all of the others.

what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
how to synthesize TiO2 nanoparticles by chemical methods
Zubear
what's the program
Jordan
?
Jordan
what chemical
Jordan
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Berger describes sociologists as concerned with
Got questions? Join the online conversation and get instant answers!


text:
## Quentin put an amount of money into each of two new investments, $$X$$ and $$Y.$$ Investment $$X$$ pays $$x\%$$ simple

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### Quentin put an amount of money into each of two new investments, $$X$$ and $$Y.$$ Investment $$X$$ pays $$x\%$$ simple

by M7MBA » Sun Sep 19, 2021 11:47 am

00:00

A

B

C

D

E

## Global Stats

Quentin put an amount of money into each of two new investments, $$X$$ and $$Y.$$ Investment $$X$$ pays $$x\%$$ simple annual interest and investment $$Y$$ pays $$y\%$$ simple annual interest. After $$1$$ year, the average (arithmetic mean) interest earned by investments $$X$$ and $$Y$$ was $$7.6\%.$$ If $$x$$ and $$y$$ are consecutive positive integers, in that order, what fraction of the total amount was put in investment $$X?$$

A. $$\dfrac15$$

B. $$\dfrac13$$

C. $$\dfrac25$$

D. $$\dfrac35$$

E. $$\dfrac23$$

Answer: C

Source: GMAT Club Tests

• Page 1 of 1


text:
# pvlib.clearsky.haurwitz¶

pvlib.clearsky.haurwitz(apparent_zenith)[source]

Determine clear sky GHI using the Haurwitz model.

Implements the Haurwitz clear sky model for global horizontal irradiance (GHI) as presented in [1, 2]. A report on clear sky models found the Haurwitz model to have the best performance in terms of average monthly error among models which require only zenith angle [3].

Parameters

apparent_zenith (Series) – The apparent (refraction corrected) sun zenith angle in degrees.

Returns

ghi (DataFrame) – The modeled global horizonal irradiance in W/m^2 provided by the Haurwitz clear-sky model.

References

1

B. Haurwitz, “Insolation in Relation to Cloudiness and Cloud Density,” Journal of Meteorology, vol. 2, pp. 154-166, 1945.

2

B. Haurwitz, “Insolation in Relation to Cloud Type,” Journal of Meteorology, vol. 3, pp. 123-124, 1946.

3

M. Reno, C. Hansen, and J. Stein, “Global Horizontal Irradiance Clear Sky Models: Implementation and Analysis”, Sandia National Laboratories, SAND2012-2389, 2012.


text:
Standing in the hall of fame

For how many positive integral values of $$n$$ does $$n!$$ end with precisely 23 trailing zeros?

×


text: 3. The simplest way to do that is to find the slope of the two lines. If two lines, with slopes $m_1$ and $m_2$, are pependicular then $m_1m_2= -1$.


text:
Practice Problems -   -  Use 6 Dot Entry   Switch to Nemeth Tutorial

# Lesson 5.5 Directed or Signed Numerals

## Review

There are indicators in UEB that apply to the item appearing just before or after the indicator. An item is defined as any one of the following groupings if they appear in the position affected by the indicator:

• An entire number, i.e. the initiating numeric symbol and all succeeding symbols within the numeric mode thus established (which would include any interior decimal points, commas, separator spaces, or simple numeric fraction lines).
• An entire general fraction, enclosed in fraction indicators.
• An arrow.
• An arbitrary shape.
• Any expression enclosed in matching pairs of round parentheses, square brackets or curly braces.
• Any expression enclosed in the braille grouping indicators.
• If none of the foregoing apply, the item is simply the next individual symbol.

## Explanation

The symbols for plus and minus are also used to indicate directed or signed values. In most cases there is no distinction between a plus or minus sign used as a sign of operation or used to indicate a directed numeral or symbol. A directed numeral designates a value as positive by using the plus sign, or negative by using the minus sign before the numeral or symbol. Usually if a number is positive, the plus sign will not be shown. However, in some cases the plus sign is shown. The plus and minus signs may also be in the superscript position. The level change up indicator places the next item in the superscript position. Grade 1 indicators must be used with the level change up indicator unless grade 1 mode has already been established.

### Example 1

${}^{+}3$
⠰⠔⠐⠖⠼⠉

### Example 2

$+3$
⠐⠖⠼⠉

### Example 3

${}^{-}3$
⠰⠔⠐⠤⠼⠉

### Example 4

$-3$
⠐⠤⠼⠉

### Example 5

${}^{-}2,\phantom{\rule{.2em}{0ex}}{}^{+}8$
⠰⠔⠐⠤⠼⠃⠂⠀⠰⠔⠐⠖⠼⠓

### Example 6

$-2,\phantom{\rule{.2em}{0ex}}+8$
⠐⠤⠼⠃⠂⠀⠐⠖⠼⠓

### Example 7

$-2+\left(+7\right)$
⠐⠤⠼⠃⠐⠖⠐⠣⠐⠖⠼⠛⠐⠜

A grade 1 symbol indicator is used with the first level change up indicator in Example 8. The numeric indicator turns on grade 1 mode for the second level change up indicator in the expression. Grade 1 mode set by the numeric indicator is only terminated by a space, hyphen, or dash.

### Example 8

${}^{+}3+\phantom{\rule{.2em}{0ex}}{}^{-}5$
⠰⠔⠐⠖⠼⠉⠐⠖⠔⠐⠤⠼⠑


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# Homework Help: Where does one term end and the other begin?

1. Apr 5, 2005

### trigger352

I need help determining the explicit formula and writing the series in sigma notation with a specified lower limit.

In both of them, I cannot tell where one term ends and the other begins, or what to do when the sign changes.

Series #1
$$- 2 - 5 - 8 - 11 - 14 - 17 - 20$$; lower limit = $$3$$
Series #2
$$4.4 + 2.4 + 0.4 - 1.6 - 3.6 - 5.6 - 7.6$$; lower limit = $$2$$

2. Apr 6, 2005

### chroot

Staff Emeritus
Each term is a number, either positive or negative. Since this is a series, you are implicitly adding all the terms. The terms which are negative get rid of the plus sign before them, per convention.

- Warren


text:
# BGRABitmap tutorial TAChart

English (en) français (fr)

You can make beautiful charts using BGRABitmap. Add TAChartBGRA package with the project inspector.

Maybe the package will not appear in the list or will not be installed. In this case, open it manually with the menu Package, Open package file (.lpk). It must be in component\tachart. Then click on the button to install the package.

TAChart in fact has the classname TChart. To add a chart, browse the components, choose Chart tab and click on TChart icon. Click and draw a rectangle on the window to drop it.

In order to supply data, click on TRandomChartSource component icon and click once in the window to drop it. In the Object Inspector, define PointsNumber to 10, XMax to 10 and YMax to 10.

Now let's create a series that uses this source. To do that, click on your chart and in the Object Inspector, go to Series property. Click on the three dots, and add a Bar series. In its properties, go to Source property and choose RandomChartSource1. The chart should be filled with red bars.

As we are going to do some 3D, we need some depth. So set the Depth property of the series to 10.

Finally, click on the chart and set the Align property to alClient in order to fill the window.

## Decorate in normal rendering mode

Go to the Series property of your chart. Choose the bar series, and in the Object Inspector, go to the events and define OnBeforeDrawBar. Add the following code:

uses TABGRAUtils;

procedure TForm1.BeforeDrawBarHandler(ASender: TBarSeries; ACanvas: TCanvas;
const ARect: TRect; APointIndex, AStackIndex: Integer;
begin
DrawPhong3DBar(ASender, ACanvas, ARect, APointIndex);
end;

Then run the program.

Your chart should look like this:

You can also use the DrawChocolateBar function. Set PointsNumber property of RandomChartSource1 to 5. Change the code of the OnBeforeDrawBarEvent to:

uses TABGRAUtils;

procedure TForm1.BeforeDrawBarHandler(ASender: TBarSeries; ACanvas: TCanvas;
const ARect: TRect; APointIndex, AStackIndex: Integer;
begin
DrawChocolateBar(ASender, ACanvas, ARect, APointIndex, True);
end;

## Using BGRABitmap to render the whole chart

Bars are not very different with or without antialiasing. Go to the Series property and remove the bars, add a Line series, and set its Source to RandomChartSource1. Then, set its property Depth to 10 and its SeriesColor property to clRed.

In order to change the rendering engine, look for the TChartGUIConnectorBGRA component in the Chart tab. Click to drop it on the chart. Then, click on the chart background and with the object inspector, set the GUIConnector property by choosing ChartGUIConnectorBGRA1. The rendering is now done with antialiasing, no need to start the program to see:

## Drawing a chart on any Canvas

### Simple drawing

First add the TAChartBGRA package. To render your chart, you can for example use a PaintBox. In the OnPaint event, write :

uses BGRABitmap, TADrawerBGRA;

procedure TForm1.PaintBox1Paint(Sender: TObject);
var
bmp: TBGRABitmap;
id: IChartDrawer;
rp: TChartRenderingParams;
begin
bmp := TBGRABitmap.Create(PaintBox1.Width, PaintBox1.Height);
Chart1.DisableRedrawing;
try
id := TBGRABitmapDrawer.Create(bmp);
id.DoGetFontOrientation := @CanvasGetFontOrientationFunc;
rp := Chart1.RenderingParams;
Chart1.Draw(id, Rect(0, 0, PaintBox1.Width, PaintBox1.Height));
Chart1.RenderingParams := rp;
bmp.Draw(PaintBox1.Canvas, 0, 0);
finally
Chart1.EnableRedrawing;
bmp.Free;
end;
end;

To avoid flickering, consider using TBGRAVirtualScreen from BGRAControls.

### Text effects

To add text effects, you need the latest version of BGRABitmap, and the latest SVN of Lazarus. There is a FontRenderer property in each TBGRABitmap image, that you can define. You must not free the renderer, as it is automatically freed by the image.

The text effects are provided by the BGRATextFX unit that contains the TBGRATextEffectFontRenderer class. To add a golden contour to the letters and a shadow, add the following lines :

uses BGRATextFX, BGRAGradientScanner;
...
var
...
fontRenderer: TBGRATextEffectFontRenderer;
end;


text:
# Copper conducts better than Gold?

#### dor

Joined Feb 20, 2009
63
Isn't Gold "known" to conduct better?
Otherwise why the pins of chips were (in older days) coated with gold?

#### Alec_t

Joined Sep 17, 2013
11,817
Because gold doesn't oxidise.

#### OBW0549

Joined Mar 2, 2015
3,566
Isn't Gold "known" to conduct better?
No. The electrical conductivity of metals is something that has been accurately known for well over a century.

Otherwise why the pins of chips were (in older days) coated with gold?
Unlike copper, gold doesn't tarnish. That makes it useful in electrical contacts, such as in connectors. Also, the lack of oxidation makes gold easy to solder. Hence, its widespread use in electronics.

#### alfacliff

Joined Dec 13, 2013
2,458
silver is the best conductor. copper is better than gold. gold is a "noble metal" and does not oxidise. copper and silver do oxidise, which will raise their restivity.

#### GopherT

Joined Nov 23, 2012
8,012
In the following page:

The table (bottom of the article) lists resistivity of Gold, Copper and other materials.

It says that resistivity of Gold is higher than Copper..
silver is the best conductor.
For a given size of wire, true.

However, is it really fair to say a wire that weighs 10% more than copper is a "better" conductor? And costs $490.00/kg while copper costs$4.65/kg.

Going the other direction, is copper such a great conductor when an aluminum cable of half the mass can carry as much current? That means, for every $4.65 of copper, you only need$0.65 of aluminum.

The resistivity x density product is the standard for mass conductivity (all other comments so far have related to the volumetric conductivity).

It is understood that joining the aluminum to other aluminum cables can be an issue and has its own costs (otherwise aluminum PCBs would be the "gold" standard in the industry).

My point is, "best" is not always best. It depends on the situation. Power lines where weight and cost are important, aluminum is a superior choice. In corrosive situations or where soldering is an issue, gold-plated copper is good. If abrasion is an issue, solid gold is best (if the application can support the price).

Note that the three most conductive materials on a mass basis are the three most easily corroded materials - sodium, lithium and potassium (in order of resistance density product, "best" being sodium). Interestingly, in large volumes, sodium is roughly half the price of copper per kilogram. And, sodium can carry 3x the current for an equivalent mass. Therefore, a $4 cable of copper could be replaced with about$0.70 worth of sodium.

Unfortunately, sodium melts at about 100 C - suddenly a liquid conductor!

Sodium/potassium alloys exist as liquids at room temperature. Imagine Eddie currents in liquid metal? At room temperature?

Also, materials that are good electrical conductors are also good thermal conductors. Now, imagine the heat sinks that could be made with these alloys - excellent thermal conductors to draw heat away from the source, but it can also be pumped. It can also withstand 800 C while developing a vapor pressure of only 1 ATM (if kept from water or oxygen).

Good night!

#### KL7AJ

Joined Nov 4, 2008
2,229
It's just REALLY hard to solder to sodium wires.

#### GopherT

Joined Nov 23, 2012
8,012

#### alfacliff

Joined Dec 13, 2013
2,458
trying to say that just because a metal is cheaper makes it a better conductor is false. and for silver being heavier is what makes it a better conductor, conductors will be smaller to handle the same current as copper. if it were weight that makes conductors better, uranium would be a great conductor.

#### boatsman

Joined Jan 17, 2008
186
@GopherT
You forgot to mention that sodium and potassium are so reactive in air that they are normaly stored covered in mineral oil.

#### GopherT

Joined Nov 23, 2012
8,012
@GopherT
You forgot to mention that sodium and potassium are so reactive in air that they are normaly stored covered in mineral oil.
Laboratory samples or small commercial quantities, yes, in mineral oil. Commercial quantities in truckloads or rail cars, are just kept in a closed containers. Melted, and pumped to the usage point. But yes, as I said above, "the most easily corroded metals."

trying to say that just because a metal is cheaper makes it a better conductor is false.
My point seemed to fly past you. I said above,
There is no single definition of "Best". "Best" depends on the situation.

#### alfacliff

Joined Dec 13, 2013
2,458
best conductor of electricity is a standard.based on resistivity.and resistivity is based on standard size conductors. if you want to specify heat conduction, silver is best, if you want to specify resistance to corrosion, gold is better. if you want the most resistant to heat, tungsten is better, you said in the origional post, conduction.

#### BR-549

Joined Sep 22, 2013
4,938
How does graphene conduction fit in here.

#### alfacliff

Joined Dec 13, 2013
2,458
it fits when it gets published . but everything has to have standards for measurement. having the resistance of a ton of gold compared to an ounce of copper is invalid. when graphene gets to where they can fabricate it ito the standard for resistivity measurement..

#### BR-549

Joined Sep 22, 2013
4,938
ok, so your saying that until we manufacture graphene to where we can carry comparable currents, we can't really make a comparison. I guess that makes sense.

#### alfacliff

Joined Dec 13, 2013
2,458
no, I am saying that when graphene can be manufactured to the resistivity standards length and area so it can be measured, it will have just a guess as to its resistivity.
how would you compare the resistivity of 1/2 inch of #32 copper to a foot of 1/2 inch thick silver? there have to be standards of measurements. going by the silly comparisons to compare resistivity is not scientific. resistivity standards are also standards of area and length of material to be measured. equal length and area of copper and gold when compared give copper the lower resistivity at the same tempratures

#### BR-549

Joined Sep 22, 2013
4,938
What do you think I meant by "comparable currents"? I wouldn't call 1/2 " of #32 copper and 1/2 foot of silver............comparable either. I was trying to say I understand your point.

#### GopherT

Joined Nov 23, 2012
8,012
best conductor of electricity is a standard.based on resistivity.and resistivity is based on standard size conductors. if you want to specify heat conduction, silver is best, if you want to specify resistance to corrosion, gold is better. if you want the most resistant to heat, tungsten is better, you said in the origional post, conduction.

I said that another method to compare, depending on needs, is to compare RESISTIVITY x DENSITY PRODUCT


text:
# Ellipse (S.N.Dey) | Part-4 | Ex-5

###### In the previous article , we solved few solutions of Short Answer Type Questions of Ellipse Chapter of  S.N.Dey mathematics, Class 11. In this chapter, we will solve few more.

A point moves so that its distance from is times its distance from the line Show that the locus of the moving point is an ellipse whose equation you are to determine.

Solution.

Let the co-ordinates of any point be so that distance of the point from the point is

Again, distance of the point from the line is

By question,

Hence, the equation of the moving point is an ellipse and is given by

Find the equation of the ellipse whose major axis is parallel to axis and

centre is , eccentricity is and the length of latus rectum is

centre is length of major axis and the co-ordinates of foci are and

Solution (i)

The length of the latus rectum

Hence, the equation of the ellipse is given by

Solution(ii)

The length of the major axis

The distance between the foci is

Hence, the equation of the ellipse is

Find the equation of the ellipse , for which the foci are and and length of the minor axis is unit. Explain how the ellipse is reduced to a circle when its two foci coincide.

Solution.

The length of minor axis

The distance between two foci is given by

From the position of foci we can easily conclude that the ellipse will be of the form

2nd Part :

When two foci coincide, then the distance between two foci

Hence, the equation of the ellipse is reduced into

So, the equation represents a circle.

The eccentricity of an ellipse is focus is and the major axis and directrix intersect at Find the co-ordinates of the centre of the ellipse.

Solution.

Let the centre of the ellipse be Now, the focus divides the line segment at the ratio internally.

Hence, the centre of the ellipse is

The lengths of major and minor axes of an ellipse are and and their equations are and respectively. Find the equation of the ellipse.

Solution.

Since the major axis and minor axes are parallel to the -axis and -axis respectively. So, the equation of the ellipse can be taken in the form where is the centre of the ellipse.

The length of the major axis :

The length of the major axis :

Now, the centre of the ellipse is the intersection of and which is

Hence, the equation of the ellipse is

Show that the point lies on the ellipse Show further that the sum of its distances from two foci is equal to the length of its major axis.

Solution.

Since the point satisfies the equation of the ellipse, the point lies on the ellipse.

Again,

So,

Hence, the co-ordinates of the foci is

So, the sum of its distances from the two foci is

Prove that for the ellipse and are two foci of the ellipse of the ellipse and is any point on the ellipse.

Solution.

The equation of the given ellipse is i.e.,

Comparing with , we get

So, the co-ordinates of any point on the ellipse can be written as

So, the co-ordinates of the foci


text:
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# Find the roots of the following quadratic equations by factorisation √ 2x^2 +7x +5√ 2 =0 - Mathematics

Course

#### Question

Find the roots of the following quadratic equations by factorisation

sqrt2 x^2 +7x+ 5sqrt2 = 0

#### Solution 1

sqrt2 x^2 + 7x + 5sqrt2

= sqrt2 x2 + 5x + 2x + 5sqrt2

= x (sqrt2x + 5) + sqrt2(sqrt2x + 5)= (sqrt2x + 5)(x + sqrt2)

Roots of this equation are the values for which (sqrt2x + 5)(x + sqrt2) = 0

:.sqrt2x + 5 = 0 or x + sqrt2 = 0

⇒ x = (-5)/sqrt2 or x = -sqrt2

#### Solution 2

We write 7x = 5x + 2x as sqrt2x^2 xx 5sqrt2 = 10x^2 = 5x xx 2x

:.sqrt2x^2 + 7x + 5sqrt2 = 0

=>sqrt2x^2+ 5x + 2x + 5sqrt2 = 0

=> x(sqrt2x + 5) + sqrt2(sqrt2x + 5) = 0

=> (sqrt2x + 5)(x + sqrt2) = 0

=> x + sqrt2 = 0or sqrt2x + 5 = 0

=> x = -sqrt2 or x = - 5/sqrt2 = -(5sqrt2)/2

Hence the roots of the given equation are -sqrt2 and - (5sqrt2)/2

Is there an error in this question or solution?