A company runs food service concessions for sporting events throughout the count

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 the demand for the new hot dog is given approximately by

p= 7 - ln(x) when 5<x<500

where x is the number of hot dogs (in thousands) that can be sold during one game at a price of p dollars. If the company pays 1 dollar for each hot dog, how should the hot dogs be priced to maximize the profit per game?

Explanation / Answer

p = 7 - ln(x) when 5 < x < 500

Total number of dogs sold = x thousand

So, total revenue = x * p

R = x(7 - ln(x))

R = 7x - xln(x)

Company pays 1 dollar per dog

So, total cost , C = 1 * x = x

Profit = R - C

P = 7x - xln(x) - x

P = 6x - xln(x)

Deriving :

dP/dx = 6 - d/dx(xln(x))

dP/dx = 6 - x*d/dx(ln(x)) - ln(x)*d/dx(x)

dP/dx = 6 - x(1/x) - ln(x)

dP/dx = 5 - ln(x)

Since this must be maximized, dP/dx = 0

5 - ln(x) = 0

ln(x) = 5

x = e^5

Now, p = 7 - ln(x)

Plug in the value of x :

p = 7 - ln(e^5)

p = 7 - 5

p = 2

So, each dog must be priced at $2 to maximize profit ----> ANSWER

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.