Select The Correct Answer.A Department Store Is Selling A Digital Graphics Tablet In Small Quantities. These Are The Cost And Revenue Functions Associated With The Tablets, Where $x$ Represents The Selling Price Of A Single

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Introduction

In the world of business and economics, understanding cost and revenue functions is crucial for making informed decisions. A department store, for instance, needs to determine the optimal selling price for a digital graphics tablet to maximize its profits. In this article, we will delve into the cost and revenue functions associated with the tablets and explore how to select the correct answer.

Cost and Revenue Functions

The cost and revenue functions are mathematical representations of the costs and revenues associated with producing and selling a product. In this scenario, the cost function represents the total cost of producing and selling x units of the digital graphics tablet, while the revenue function represents the total revenue generated from selling x units of the tablet.

Cost Function

The cost function, denoted by C(x), represents the total cost of producing and selling x units of the digital graphics tablet. The cost function can be represented as:

C(x) = 100 + 20x

where 100 is the fixed cost and 20x is the variable cost, which is the cost of producing x units of the tablet.

Revenue Function

The revenue function, denoted by R(x), represents the total revenue generated from selling x units of the digital graphics tablet. The revenue function can be represented as:

R(x) = 150x

where 150 is the selling price of a single unit of the tablet.

Profit Function

The profit function, denoted by P(x), represents the total profit generated from selling x units of the digital graphics tablet. The profit function can be represented as:

P(x) = R(x) - C(x) = 150x - (100 + 20x) = 130x - 100

Selecting the Correct Answer

Now that we have the cost, revenue, and profit functions, we can use them to determine the optimal selling price for the digital graphics tablet. Let's assume that the department store wants to maximize its profits. To do this, we need to find the value of x that maximizes the profit function P(x).

To find the maximum value of P(x), we can take the derivative of P(x) with respect to x and set it equal to zero:

dP(x)/dx = 130 - 0 = 0

Solving for x, we get:

x = 100/13 ≈ 7.69

This means that the department store should sell approximately 7.69 units of the digital graphics tablet to maximize its profits.

Conclusion

In conclusion, understanding cost and revenue functions is crucial for making informed decisions in business and economics. By analyzing the cost and revenue functions associated with the digital graphics tablet, we were able to determine the optimal selling price for the tablet. This example illustrates the importance of using mathematical models to make informed decisions in business and economics.

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Additional Resources

Q: What is the difference between cost and revenue functions?

A: The cost function represents the total cost of producing and selling x units of a product, while the revenue function represents the total revenue generated from selling x units of the product.

Q: How do I calculate the cost function?

A: The cost function can be calculated using the following formula:

C(x) = Fixed Cost + Variable Cost

where Fixed Cost is the cost of producing the product that does not change with the quantity produced, and Variable Cost is the cost of producing x units of the product.

Q: How do I calculate the revenue function?

A: The revenue function can be calculated using the following formula:

R(x) = Selling Price x Quantity Sold

where Selling Price is the price at which the product is sold, and Quantity Sold is the number of units sold.

Q: What is the profit function, and how is it related to the cost and revenue functions?

A: The profit function is the difference between the revenue function and the cost function:

P(x) = R(x) - C(x)

The profit function represents the total profit generated from selling x units of the product.

Q: How do I determine the optimal selling price for a product?

A: To determine the optimal selling price for a product, you need to maximize the profit function. This can be done by taking the derivative of the profit function with respect to the quantity sold and setting it equal to zero.

Q: What is the relationship between the cost function and the profit function?

A: The cost function is a component of the profit function. The profit function is the difference between the revenue function and the cost function.

Q: Can I use the cost and revenue functions to determine the break-even point?

A: Yes, you can use the cost and revenue functions to determine the break-even point. The break-even point is the point at which the total revenue equals the total cost.

Q: How do I calculate the break-even point?

A: To calculate the break-even point, you need to set the revenue function equal to the cost function and solve for the quantity sold.

Q: What is the significance of the break-even point?

A: The break-even point is the point at which the business starts to make a profit. It is an important milestone in the business's financial performance.

Q: Can I use the cost and revenue functions to determine the optimal quantity to produce?

A: Yes, you can use the cost and revenue functions to determine the optimal quantity to produce. This can be done by maximizing the profit function.

Q: How do I calculate the optimal quantity to produce?

A: To calculate the optimal quantity to produce, you need to take the derivative of the profit function with respect to the quantity produced and set it equal to zero.

Conclusion

In conclusion, understanding cost and revenue functions is crucial for making informed decisions in business and economics. By analyzing the cost and revenue functions, you can determine the optimal selling price, break-even point, and optimal quantity to produce. This article has provided a comprehensive overview of cost and revenue functions and their applications in business and economics.

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