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
Mathematics · High School · Thu Feb 04 2021
Answered on
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.