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

Define the cost function

The cost of manufacturing x crayons is given by the function \[C(x) = 570 + 0.5x\]

Define the revenue function

The revenue from selling x crayons is given by the function \[R(x) = 10x\]

Set the revenue equal to the cost

To find the break-even point, set the cost function equal to the revenue function and solve for x: \[570 + 0.5x = 10x\]

Solve for x

Isolate x by first subtracting 0.5x from both sides: \[570 = 9.5x\] Next, divide both sides by 9.5 to find x: \[x = \frac{570}{9.5} = 60\]

Verify the solution

Calculate the cost and revenue at x = 60: \[C(60) = 570 + 0.5(60) = 600\] \[R(60)= 10(60)= 600\] Since the cost and revenue are equal, the solution is verified.

Key concepts

Cost Function

The cost function represents how much it costs to produce a certain number of items.
In this exercise, the cost function for making crayons is given by \[ C(x) = 570 + 0.5x \] where \(x\) stands for the number of crayons. The fixed cost is 570 cents, meaning it costs 570 cents to start the manufacturing.
This stays constant regardless of the number of crayons made. The term \(0.5x\) shows the variable cost, or how much more it costs for each additional crayon.
Here, it adds 0.5 cents for each crayon.
By distinguishing these two parts, fixed and variable costs, you can understand better how changes in production impact total costs.

Revenue Function

The revenue function calculates the money earned from selling a certain number of items. In our case, the revenue function for selling crayons is given by \[ R(x) = 10x \] Here, \(x\) is the number of crayons sold, and 10 is the price in cents for each crayon.
This function shows a direct relationship between the number of units sold and the revenue generated.
The more crayons sold, the higher the revenue.
Understanding the revenue function helps to see how sales impact earnings directly.

Equation Solving

To find the break-even point where revenue equals cost, you need to solve an equation. Here, set the cost function equal to the revenue function:\[ 570 + 0.5x = 10x \] To solve this, first isolate \(x\) by removing \(0.5x\) from both sides: \[ 570 = 9.5x \] Then, divide both sides by 9.5: \[ x = \frac{570}{9.5} = 60 \] This step-by-step approach shows that selling 60 crayons will bring in enough revenue to cover the manufacturing cost.
Verifying this involves plugging \(x = 60\) back into both the cost and revenue functions: \[ C(60) = 570 + 0.5(60) = 600 \] \[ R(60) = 10(60) = 600 \] Both sides match, confirming the solution.
Equation solving is crucial to find such break-even points, balancing costs and revenues over different scenarios.