Select The Correct Answer.These Are The Cost And Revenue Functions For A Product Line Of Cat Food Sold In 7-pound Bags At A Single Pet Store:$\[ \begin{array}{l} R(x)=700x-11.3x^2 \\ C(x)=8,068-34.25x \end{array} \\]Based On These Functions,

Economic Analysis of Cat Food Sales: Understanding Cost and Revenue Functions

In the world of economics, understanding the relationship between cost and revenue is crucial for businesses to make informed decisions. The cost and revenue functions are mathematical representations of the expenses and income generated by a company's products or services. In this article, we will delve into the cost and revenue functions of a product line of cat food sold in 7-pound bags at a single pet store. We will analyze the given functions and determine the correct answer to a series of questions.

The cost and revenue functions for the cat food product line are given by:

$$\begin{array}{l} R(x)=700x-11.3x^2 \\ C(x)=8,068-34.25x \end{array}$$

where $R(x)$ represents the revenue function and $C(x)$ represents the cost function. The variable $x$ represents the number of 7-pound bags of cat food sold.

Revenue Function

The revenue function $R(x)$ is a quadratic function that represents the total revenue generated by selling $x$ number of 7-pound bags of cat food. The function is given by:

$$R(x)=700x-11.3x^2$$

To understand the behavior of the revenue function, we can analyze its graph. The graph of a quadratic function is a parabola that opens downwards or upwards. In this case, the revenue function opens downwards, indicating that the revenue decreases as the number of bags sold increases.

Cost Function

The cost function $C(x)$ is a linear function that represents the total cost incurred by selling $x$ number of 7-pound bags of cat food. The function is given by:

$$C(x)=8,068-34.25x$$

The cost function represents the fixed and variable costs incurred by the company. The fixed cost is $8,068, which is the initial cost incurred by the company to produce and sell the cat food. The variable cost is $34.25x, which represents the cost incurred by the company to produce and sell each additional bag of cat food.

Marginal Revenue and Marginal Cost

The marginal revenue and marginal cost are the rates of change of the revenue and cost functions, respectively. The marginal revenue is the additional revenue generated by selling one more unit of the product, while the marginal cost is the additional cost incurred by producing and selling one more unit of the product.

The marginal revenue function is given by:

$$MR(x)=\frac{dR(x)}{dx}=700-22.6x$$

The marginal cost function is given by:

$$MC(x)=\frac{dC(x)}{dx}=-34.25$$

Profit Function

The profit function is given by:

$$\pi(x)=R(x)-C(x)$$

Substituting the revenue and cost functions, we get:

$$\pi(x)=(700x-11.3x^2)-(8,068-34.25x)$$

Simplifying the expression, we get:

$$\pi(x)=728.25x-11.3x^2-8,068$$

Optimization

To maximize the 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 and setting it equal to zero.

$$\frac{d\pi(x)}{dx}=728.25-22.6x=0$$

Solving for $x$, we get:

$$x=\frac{728.25}{22.6}=32.25$$

Since we cannot sell a fraction of a bag of cat food, we round down to the nearest whole number. Therefore, the optimal number of bags to sell is 32.

In conclusion, the cost and revenue functions for the cat food product line are given by:

$$\begin{array}{l} R(x)=700x-11.3x^2 \\ C(x)=8,068-34.25x \end{array}$$

We analyzed the revenue and cost functions, and determined the marginal revenue and marginal cost functions. We also derived the profit function and optimized it to find the optimal number of bags to sell. The optimal number of bags to sell is 32.

The final answer is 32.
Economic Analysis of Cat Food Sales: Understanding Cost and Revenue Functions

In the previous article, we analyzed the cost and revenue functions for a product line of cat food sold in 7-pound bags at a single pet store. We derived the profit function and optimized it to find the optimal number of bags to sell. In this article, we will answer some frequently asked questions related to the cost and revenue functions.

Q: What is the revenue function?

A: The revenue function is a quadratic function that represents the total revenue generated by selling x number of 7-pound bags of cat food. The function is given by:

$$R(x)=700x-11.3x^2$$

Q: What is the cost function?

A: The cost function is a linear function that represents the total cost incurred by selling x number of 7-pound bags of cat food. The function is given by:

$$C(x)=8,068-34.25x$$

Q: What is the marginal revenue function?

A: The marginal revenue function is the rate of change of the revenue function. It represents the additional revenue generated by selling one more unit of the product. The marginal revenue function is given by:

$$MR(x)=\frac{dR(x)}{dx}=700-22.6x$$

Q: What is the marginal cost function?

A: The marginal cost function is the rate of change of the cost function. It represents the additional cost incurred by producing and selling one more unit of the product. The marginal cost function is given by:

$$MC(x)=\frac{dC(x)}{dx}=-34.25$$

Q: How do I maximize the profit?

A: To maximize the profit, you need to find the value of x that maximizes the profit function. You can do this by taking the derivative of the profit function and setting it equal to zero.

$$\frac{d\pi(x)}{dx}=728.25-22.6x=0$$

Solving for x, you get:

$$x=\frac{728.25}{22.6}=32.25$$

Since you cannot sell a fraction of a bag of cat food, you round down to the nearest whole number. Therefore, the optimal number of bags to sell is 32.

Q: What is the optimal number of bags to sell?

A: The optimal number of bags to sell is 32.

Q: What is the maximum profit?

A: To find the maximum profit, you need to substitute the optimal number of bags to sell into the profit function.

$$\pi(x)=728.25x-11.3x^2-8,068$$

Substituting x = 32, you get:

$$\pi(32)=728.25(32)-11.3(32)^2-8,068$$

Simplifying the expression, you get:

$$\pi(32)=23,296-11,488-8,068$$

$$\pi(32)=3,740$$

Therefore, the maximum profit is $3,740.

In conclusion, we have answered some frequently asked questions related to the cost and revenue functions. We have derived the marginal revenue and marginal cost functions, and optimized the profit function to find the optimal number of bags to sell. The optimal number of bags to sell is 32, and the maximum profit is $3,740.

The final answer is 32.