Problem 13 - 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. $$ \begin{array}{l} {C(x)=40 x+22,500} \\ {R(x)=85 x} \end{array} $$

Step-by-Step Solution

Answer

Profit function: \(P(x) = 45x - 22,500\). Break-even point at \(x = 500\).

Step 1 - Define Total-Profit Function

The total-profit function, denoted as \(P(x)\), is defined as the difference between the total revenue \(R(x)\) and the total cost \(C(x)\). Mathematically, \(P(x) = R(x) - C(x)\).

Step 2 - Calculate Total-Profit Function

Given \(C(x) = 40x + 22,500\) and \(R(x) = 85x\), substitute these into the total-profit function: \[P(x) = 85x - (40x + 22,500) = 85x - 40x - 22,500 = 45x - 22,500\]

Step 3 - Define Break-Even Point

The break-even point is where the total-profit function equals zero, meaning \(P(x) = 0\).

Step 4 - Solve for Break-Even Point

Set the profit function to zero and solve for \(x\): \[0 = 45x - 22,500\] \[45x = 22,500\] \[x = \frac{22,500}{45} = 500\]

Key Concepts

Break-Even Point

The break-even point is where your business doesn't make a profit or a loss. In other words, it's the point where the total revenue matches the total cost. Understanding this concept helps determine how much you need to sell to cover all your expenses.

Think about it this way: if you sell less than the break-even point, your costs are higher than your sales, leading to a loss. If you sell more, you're making a profit.

The break-even point is crucial in planning and decision-making, especially for new businesses. It helps you to set sales targets and understand the minimum performance needed to avoid losses.

In our exercise, we calculated the break-even point by setting the total-profit function to zero:

Total Revenue

Total revenue is the total amount of money generated from selling products or services. It's calculated by multiplying the price per unit by the number of units sold.

In our example, the total revenue function is given as: R(x) = 85x.

Here, '85' represents the price per unit, and 'x' represents the number of units sold. Total revenue is an essential metric because it helps businesses understand how well they are performing in terms of sales.

Knowing your total revenue can help set pricing strategies and sales goals. The higher the total revenue, the better your sales performance, but remember, this doesn't account for costs yet.

Total Cost

Total cost is the sum of all expenses a business incurs to produce and sell its products or services. It includes both fixed costs and variable costs.

In our exercise, the total cost function is given as: C(x) = 40x + 22,500.

Here, '40x' represents the variable costs that change with the number of units produced, and '22,500' represents the fixed costs that remain constant regardless of the number of units produced.

Understanding your total cost helps you price your products appropriately and manage your expenses effectively. Lowering total costs without affecting quality can lead to improved profitability.

Profit Calculation

Profit calculation is essential for understanding how much money a business is making after covering all its costs. The profit function shows the difference between total revenue and total cost.

In our exercise, we calculated the profit function as: P(x) = 85x - (40x + 22,500) = 45x - 22,500.

Here, '45x' represents the profit per unit sold, and '22,500' is subtracted to account for the fixed costs. Profit calculation helps determine the financial health of your business. Positive profit indicates that your business is doing well, while a negative profit suggests areas for improvement.

Calculating profit regularly can guide business decisions, such as pricing strategies, cost management, and potential investments in growth.