Question

A carpet cleaning business has 100 customers when they charge $125 for a cleaning. Research shows that every $5 reduction in price attracts another 20 customers. What price should the business implement to maximize its revenue?
a $75
b $100
c $120
d $90

Answer 1

The price that will maximize its revenue is $75

Answer 2

Since a $5 decrease in price increases customers by 20 we can say that we have two points:

(125,100) and (120,120), from these we can find the slope or rate of change of customers as a function of price...

m=20/-5

m=-4

m=-4

c(p)=-4p+b, now we can use (125,100) to solve for b

100=-4(125)+b

100=-500+b

600=b, so our number of customers as a function of price is:

c(p)=600-4p

Revenue will simply be the number of customers times the price charged per customer...or p*c(p):

r(p)=600p-4p^2

We can find price that creates maximum revenue by finding when the derivative is equal to zero...

dr/dp=600-8p

dr/dp=0 only when

0=600-8p

8p=600

p=75

So the price that maximizes revenue is $75.