Question

The cost, in cents, of manufacturing \(x\) crayons is \(570+0.5 x\). The crayons sell for 10 cents each. What is the minimum number of crayons that need to be sold so that the revenue received recoups the manufacturing cost? a. 50 b. 57 c. 60 d. 61 e. 95

Short answer

60

Step-by-step solution

- Understand the Problem

The task is to determine the minimum number of crayons, represented by the variable \(x\), that must be sold such that the total revenue equals the manufacturing cost.

- Define the Functions

Define the manufacturing cost function and the revenue function:1. The manufacturing cost in cents is given by \(C(x) = 570 + 0.5x\).2. The revenue function when selling each crayon for 10 cents is \(R(x) = 10x\).

- Set Up the Equation

Set the revenue function equal to the manufacturing cost function to find the break-even point: \[ 10x = 570 + 0.5x \]

- Solve for \(x\)

Solve the equation for \(x\) by isolating \(x\) on one side:\[ 10x - 0.5x = 570 \]\[ 9.5x = 570 \]\[ x = \frac{570}{9.5} \]\[ x \approx 60 \]

- Determine the Answer

Since \(x\) must be a whole number (you can't sell a fraction of a crayon), and 60 is an exact solution, verify that 60 satisfies both the manufacturing cost and revenue equations.

Key concepts

Revenue Function

In this problem, the revenue function represents the total income generated from selling a certain number of crayons. To create this function, note the selling price per crayon: each crayon sells for 10 cents.

Thus, if you sell \( x \) crayons, the revenue generated can be expressed as:

\[ R(x) = 10x \]

The revenue function is crucial in break-even analysis because it helps you determine the income generated from sales. When you set the revenue function equal to the manufacturing cost function, you can find the break-even point — where total revenue equals total costs. Understanding the revenue function is the first step in solving such problems.

Manufacturing Cost Function

The manufacturing cost function represents all costs involved in producing a certain number of crayons. In this scenario, a fixed cost is combined with a variable cost per crayon.

Given in the problem, the cost to produce \( x \) crayons is expressed as:

\( C(x) = 570 + 0.5x \)

Here, 570 cents represent the fixed cost, which doesn't change regardless of the number of crayons produced, while 0.5 cents per crayon represent the variable cost, which varies with production volume.

Understanding these components is essential to set up the equations properly for finding the break-even point. Fixed costs remain constant, while variable costs depend on the number of units produced.

Solving Equations

After defining both the revenue and manufacturing cost functions, the next step is solving the equation that equates the revenue to the manufacturing costs. This process determines the break-even point, where revenue covers all costs.

Start by setting the revenue function equal to the manufacturing cost function:

\( 10x = 570 + 0.5x \)

To isolate \( x \), first subtract 0.5x from both sides:

\( 10x - 0.5x = 570 \)

Simplify the left side:

\( 9.5x = 570 \)

Finally, solve for \( x \) by dividing both sides by 9.5:

\( x = \frac{570}{9.5} \)

Thus, \( x \approx 60 \).

To confirm correctness, you verify that if 60 crayons are sold, the revenue equals the manufacturing cost. This validation confirms the break-even point and ensures the solution is correct. Hence, 60 crayons must be sold to recoup the manufacturing costs.