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=6−ln(x),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?
Write a formula for the profit as a function of the number x of hot dogs sold.
Profit =(Value of x)
Number of hot dogs that maximizes profit:
profit=x(p-1)
profit=x(6-lnx)
profit=6x-xlnx
dprofit/dx=0=6-lnx-x/x
lnx=6-2=5
x=e^5
x=148 *thousand)
To find the profit per game, we need to consider the cost of manufacturing the hot dogs and the revenue generated from selling them.
The cost of manufacturing the hot dogs can be calculated by multiplying the number of hot dogs sold (x) by the cost per hot dog (1 dollar). Therefore, the cost function is given by:
Cost = 1x = x
The revenue generated from selling the hot dogs can be calculated by multiplying the number of hot dogs sold (x) by the price per hot dog (p) obtained from the demand equation: p = 6 - ln(x). Therefore, the revenue function is given by:
Revenue = px = (6 - ln(x))x = 6x - xln(x)
Now, the profit per game is calculated by subtracting the cost from the revenue:
Profit = Revenue - Cost = (6x - xln(x)) - x
To find the number of hot dogs that maximizes profit, we need to find the value of x that maximizes the profit function. We can do this by taking the derivative of the profit function with respect to x, setting it equal to zero, and solving for x.
d(Profit)/dx = 6 - ln(x) - 1 = 5 - ln(x) = 0
Now, solve for x:
ln(x) = 5
x = e^5
Therefore, the number of hot dogs that maximizes profit is approximately e^5 (e is Euler's number, approximately 2.71828).
To determine how the hot dogs should be priced to maximize profit, substitute the value of x into the demand equation to find the corresponding price:
p = 6 - ln(x)
p = 6 - ln(e^5)
p = 6 - 5
p = 1
Therefore, the hot dogs should be priced at 1 dollar to maximize the profit per game.