### Ball Bearings

If a is the radius of the axle, b the radius of each ball-bearing, and c the radius of the hub, why does the number of ball bearings n determine the ratio c/a? Find a formula for c/a in terms of n.

### Overarch 2

Bricks are 20cm long and 10cm high. How high could an arch be built without mortar on a flat horizontal surface, to overhang by 1 metre? How big an overhang is it possible to make like this?

### Cushion Ball

The shortest path between any two points on a snooker table is the straight line between them but what if the ball must bounce off one wall, or 2 walls, or 3 walls?

# More Realistic Electric Kettle

##### Stage: 4 and 5 Challenge Level:

An electric kettle is a quite simple household appliance which is used everyday to convert electrical energy to heat energy. The diagram shows a circuit which provides a simple model for what happens in the electric kettle.

The kettle diagram: V is the voltage of the power supply (a power supply in UK has 240 volts (V)), R is the resistance of a heating element measured in ohms ($\Omega$), I is a current flowing around the circuit in amps (A).

The experiment was carried out to investigate the relationship between the resistance of heating element and the temperature which settle down after a while. Results are shown in the table.

Resistance R/$\Omega$  Temperature T/°C
65                     97
100                    70
150                    53
200                    45
250                    40

1. Why does the temperature settle down after a while?
2. Plot the graph of data for the temperature against the resistance.
3. Could you recongise the function? Using points (100, 70) and (250, 40) find the equation of this function. (Hint: the function is of the form T = T0 + C/R where T0 and C are constants.)
4. Find the temperature of the room where the experiment was conducted.
5. Does this equation is suitable for all values of resistance?

In practise we know that for an electric kettle to boil water takes 3-4 minutes. Could you

investigate how the time needed to boil water is related with the amount of water which are you heating?

Extension: the power of the kettle is given by the formula where R is the resistance of the heating element, r is the internal resistance and V is the voltage of power supply.  $$P = \frac{RV^2}{(r+R)^2}$$ Suppose it is given that V = 240 volts, r = 5 ohms. Find R such that we get the maximum power.

Try to deduce generally the value of R for which the power is the biggest if the internal resistance is r.

An old electric kettle picture is taken from http://www.sciencemuseum.org.uk/images/I059/10325939.aspx

The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice. More information on many of our other activities can be found here.
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