Calculus Word Problem

Question

A company runs food service concessions for sporting events throughout the country. Their marketing research department chose a particular football stadium to test market a new jumbo hot dog. It was found that in order to sell  hot dogs (in thousands), the price (in dollars) should be set at

p(x)= 5-ln(x), 5<x<500,

If the company pays 1 dollar for each hot dog, how should the hot dogs be priced to maximize the profit per game?

Answer 1

Hi Madeline,

According to the question, you are asked to find the price where profit is maximized. And the profit is calculated by x (number of hot dogs selling) multiplied by the price minus x ( 1 dollar cost per hot dog).

Revenue = x * p(x) = 5x - xln(x)

Profit = x*p(x)-x= 4x - xln(x)

We need to take the derivative of the profit function to find the x value where profit is maximized.

So derivative of 4x-xln(x) is dp/dx = 4- (x*1/x+lnx)= 4-1-lnx = 3 - lnx

We set the first derivative equals to 0 to find the maximized point.

3 - lnx = 0, which means lnx = 3.

The rest is simple, plug lnx = 3 back into p(x)= 5-ln(x) to find the price, p(x) = 5 - 3 = 2

So the answer is 2. Hope it helps.