Demand function

Sep 2008
222
0
the problem deals with applications of derivatives. the problem is: until recently hamburgers at the city sports arena costs $4 each the food concessionaire sold an average of 10,000 hamburgers on a game night. when the price was raised to $4.40, hamburger sales dropped off to an average of 8000 per night.

a) find the price of a hamburger that will maximize the nightly hamburger revenue.

I know that in order to do that, we have to use the point slope formula to find the demand function. I got p-4=[(4-4.4)/(10,000-8000)] [(x-10000).

I don't get why the answer key has p-2=[(2-2.4)/(10,000-8000)] [(x-10000) and still got the correct answer?
 
Aug 2007
3,171
860
USA
Where did you use the derivative?

x = Price
Demand(x) = a + bx -- We're assuming a linear model.
Revenue(x) = x*Demand(x) = ax + bx^2

Now we have clues:

Demand(4) = a + b(4) = 10000
Demand(4.4) = a + b(4.4)= 8000

I get a = 30000 and b = -5000

Revenue(x) = 30000*x - 5000*x^2

dRevenue/dx = 30000 - 10000*x

Find zero: 30000 - 10000*x ==> x = 3
Revenue(3) = 45000

Write down definitions. Write clearly and completely. Make sure at least YOU can follow your work.
 

HallsofIvy

MHF Helper
Apr 2005
20,249
7,909
TKHunny, I don't believe that was the question. The OP was simply asking, "why did the answer key use 'p -2= =[(2-2.4)/(10,000-8000)] [(x-10000)' rather than 'p- 4= [(4- 4.4)/(10,000- 8000)][(x- 10000)' and how do both of those give the same answer?"

fabxx, all I can say is that there must have been a typo. For the given problem, yours is the correct formula and they give different answers. Using the answer key's demand function, the price that gives maximum revenue would, in fact, be $2.00.
 
Aug 2007
3,171
860
USA
Making up my own questions again, eh? (Doh)