Question

The stadium vending company finds that sales of hot dogs average 32,000 hot dogs per game when the hot dogs sell for $2.50 each. For each 50 cent increase in the price, the sales per game drop by 5000 hot dogs. What price per hot dog should the vending company charge to realize the maximum revenue?

Answer 1

0.35 cents

Explanation:

Let n be the numbers of times the vending company increases in 0.50 dollars the price of its hot dogs, then the income function per game would be

I(n) = (32,000 - 5,000n)(2.50 + 0.50n)

and we have to find the value of n that maximizes the income.

We can write I(n) as a polynomial

`\bf I(n)=-2,500n^2+3,500n+80,000`

Taking the derivative

I'(n) = -5,000n+3,500

I'(n) = 0 when 5,000n=3,500

So n=3,500/5,000 = 0.7

and n=0.7 is a critical point of I

Taking the second derivative

I''(n) = -5,000

Since I''(0.7) = -5,000 < 0 we see n=0.7 is a maximum of I(n)

Hence, the vending company must increase the price 0.7 times 0.5 dollars, that is to say, 0.35 cents.