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),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 gam

Answer 1

Answer:

The company should sell 148 hot dogs, at a price of $2 each

Explanation:

Profit Maximization

We need to find the number of jumbo hot dogs (x) that need to be sold during one game to maximize the profits of the company. There is some information we need to manage to achieve this maximization.

First, we have the equation of the demand is given by

$p(x)=7-lnx$

where x is limited to the interval [5,500]

The revenue function is given by the product of the price by the number of jumbo hot dogs sold

$R(x)=x.p(x)=7x-x.lnx$

We also know the company pays 1 dollar for each hot dog, thus the profit function is

$P(x)=R(x)-1.x=7x-x.lnx-x=6x-x.lnx$

To maximize the profit, we set its derivative to 0

$P'(x)=(6x-x.lnx)'=6x'-(x.lnx)'$

Applying the derivative of a product

$P'(x)=6-(x'.lnx+x(1/x))=6-lnx-1=5-lnx$

Equating to 0

$5-lnx=0$

Solving for x

$x=e^5=148.41$

The solution lies on the given interval, so it's a valid value

We choose the closest integer because x cannot be decimal

The price should be

$p(148)=7-148.ln(148)=2$

The company should sell 148 hot dogs, at a price of $2 each

Answer 2

To maximize profits, you will need to sell 148 hot dogs for at least $2 each.

We can arrive at this answer as follows:

• First, we must interpret the demand equation. This will be done with the formula:

$p(x)=7-lnx$

• We will also need to consider the recipe function, which will be obtained with the following formula:

$R(x)=x*p(x)=7x-xlnx$

With this equation, we will know the relationship between the price and quantity of hot dogs that must be sold.

• It is now possible to relate the two equations, taking into account that the company will pay one dollar per hot dog sold. This will be done as follows:

$P(x)=R(x) -1x=7x-xlnx-x=6x-xlnx$

• By simplifying this equation we will have:

$P(x)= (6x-slnx)'=6x'-(xlnx)'$

• With the derivation of the product, we will have:

$P'(x)=6-(x'lnx+x*\frac{1}{x} )=6-lnx-1=5-lnx$

• With this equation, we need to find the value of x. This will be done as follows:

$x=e^5=148.41$

• Disregarding the decimals and applying this value to the demand equation, we will have:

$p(148)=7-148*ln8(148)=2$

Therefore, we can conclude that to optimize profits, it is necessary to sell at least 148 hot dogs for $2 each.

More information:

https://brainly.com/question/13353440?referrer=searchResults