Problem 14 - Elementary and Intermediate Algebra

Question

For each of the following pairs of total-cost and total revenue functions, find (a) the total-profit function and (b) the break-even point. $$ \begin{aligned} &C(x)=20 x+10,000\\\ &R(x)=100 x \end{aligned} $$

Step-by-Step Solution

Answer

The total-profit function is \( P(x) = 80x - 10,000 \) and the break-even point is 125 units.

Step 1: Understand the total-cost and total-revenue functions

The total-cost function is given as \( C(x) = 20x + 10,000 \) and the total-revenue function is given as \( R(x) = 100x \), where \( x \) represents the number of units produced and sold.

Step 2: Find the total-profit function

The total-profit function \( P(x) \) is found by subtracting the total-cost function from the total-revenue function. \[ P(x) = R(x) - C(x) \] Substitute the given functions: \[ P(x) = 100x - (20x + 10,000) \] Simplify the equation: \[ P(x) = 100x - 20x - 10,000 \] \[ P(x) = 80x - 10,000 \]

Step 3: Find the break-even point

The break-even point occurs when total profit is zero, \( P(x) = 0 \). Set the profit function equal to zero and solve for \( x \): \[ 80x - 10,000 = 0 \] Add 10,000 to both sides: \[ 80x = 10,000 \] Divide both sides by 80: \[ x = \frac{10,000}{80} \] \[ x = 125 \] Therefore, the break-even point is when 125 units are produced and sold.

Key Concepts

total-cost function

The total-cost function represents the overall cost incurred by a company to produce and sell a certain number of units. In this exercise, the total-cost function is given by \( C(x) = 20x + 10,000 \). Here, \( 20x \) is the variable cost, which means it depends on the number of units produced, \( x \). The term \( 10,000 \) is the fixed cost, which remains constant regardless of the number of units. Fixed costs could include expenses like rent, salaries, or utilities. So, if you produce more units, your variable costs increase, but your fixed costs stay the same. This function helps you understand how costs behave as production scales.

total-revenue function

The total-revenue function indicates the total income a company earns from selling a certain number of units. In this case, the total-revenue function is \( R(x) = 100x \). Here, \( 100 \) represents the price per unit, and \( x \) stands for the number of units sold. Thus, the revenue increases linearly as more units are sold. This is useful for predicting how much money your sales will bring in as you ramp up production. If you sell more units, your revenue will increase proportionally.

break-even point

The break-even point is a crucial concept for any business, marking the threshold where total revenues exactly match total costs, meaning the company does not make a profit but also does not incur a loss. To find the break-even point from the given functions, you need to set the total-profit function to zero. The total-profit function is derived by subtracting the total-cost function from the total-revenue function: \( P(x) = R(x) - C(x) \). For this exercise:

• Total-Cost: \( C(x) = 20x + 10,000 \)
• Total-Revenue: \( R(x) = 100x \)

Substituting these into the profit function gives: \[ P(x) = 100x - (20x + 10,000) = 80x - 10,000 \] To find the break-even point, set \( P(x) \) to zero and solve for \( x \):

• \( 80x - 10,000 = 0 \)
• Add 10,000 to both sides: \( 80x = 10,000 \)
• Divide both sides by 80: \( x = \frac{10,000}{80} = 125 \)

So, the break-even point is reached when 125 units are produced and sold. This indicates that at this point, total revenue covers total costs and any units sold beyond this will generate profit.