# Thread:

1. ## Help with finding rate of increase using product/quotient rule

I've been having trouble with a particular problem from my Calculus book in the Product and Quotient rules chapter.

I was hoping someone on this forum could help me out. Thanks in advance!

Question:

Suppose the price of an object is \$14 and 12,000 units are sold. The company wants to increase the quantity sold by 1200 units per year, while increasing the revenue by \$20,000 per year. At what rate would the price have to be increased to reach these goals?

2. Originally Posted by elong
I've been having trouble with a particular problem from my Calculus book in the Product and Quotient rules chapter.

I was hoping someone on this forum could help me out. Thanks in advance!

Question:

Suppose the price of an object is \$14 and 12,000 units are sold. The company wants to increase the quantity sold by 1200 units per year, while increasing the revenue by \$20,000 per year. At what rate would the price have to be increased to reach these goals?
If x is the price of each object, than the price, P, times x is the revenue. So $R=Px$. For the first part, P=\$14 and x=12,000 so $R=14 \times 12,000=168,000$. Now if the amount sold has to been increased by 1,200 units per year this means that the new total objects sold is 12,000+1,200=13,200. Follow?

So now we know the units sold and the desired revenue, but not the price. Let's set up the equation again: $R=Px$ so $20,000=P(13,200)$. Now solve for P and solve for how much this new price has increased from the old.

edit: Your problem might be a little different than this. I don't know if the first line means those products were sold once or that's the yearly amount.

3. Thank you. I do follow what you are saying and it does seem logical, but what is throwing me off is the fact that I think we are suppose to either use the product or quotient rule to solve this problem.

The following is a similar question that I solved correctly.

Ques:

Suppose the price of an object is \$20 and 20,000 units are sold. If the price increases at a rate of \$1.25 per year and the quantity sold increases at a rate of 2000 per year, at what rate will revenue increase?

Answer:

R' = Q'P + QP' (Product Rule)

At a certain moment of time (call it t0) we are given P(t0) = 20.

Q(t0) = 20,000 (items)
P'(t0) = 1.25
Q'(t0) = 2,000 (items/year)

--> R'(t0) = 2,000(20) + (20,000)1.25)

= 65,000 \$ per year
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