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

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

We can arrive at this answer as follows:

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

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

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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

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

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

To maximize the profit, we set its derivative to 0

Applying the derivative of a product

Equating to 0

Solving for x

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

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

reducing the scale of the market for a new product.