Problem 58 - Contemporary Precalculus

Question

The cost \(C\) of making \(x\) cases of pet food is given by \(C=100 x+2600 .\) Each case of 100 boxes sells for \(\$ 120\) (a) Find an equation that expresses the revenue from selling \(x\) cases. (b) How many cases must be sold for the company to break even?

Step-by-Step Solution

Answer

Answer: The company must sell 130 cases of pet food to break even.

Step 1: Understand Cost and Revenue functions

The cost function (\(C\)) and the revenue function (\(R\)) are both related to the total amount produced (\(x\)). The cost function expresses how much it costs to produce \(x\) cases, and the revenue function expresses how much we get by selling \(x\) cases.

Step 2: Find the equation for the Revenue function (R)

Since each case of 100 boxes sells for \(\$120\), if we sell \(x\) cases, the revenue would be the number of cases multiplied by the revenue per case. So, \(R = 120 \cdot x\).

Step 3: Calculate the Break-Even Point

To find the break-even point, we'll look for when the revenue equals the cost, or \(R = C\). Using the equations we've found: \(120x = 100x + 2600\)

Step 4: Solve for x

To find the value of \(x\) that will make the company breakeven, we should solve the equation \(120x = 100x + 2600\). First, let's express the equation in a more standard way: \(20x = 2600\) Now, to isolate x, we simply divide both sides of the equation by 20: \(x = \frac{2600}{20}\)

Step 5: Calculate the number of cases to break even

Now we can compute the value of x by dividing 2600 by 20: \(x = 130\) So, the company must sell 130 cases of the pet food to break even.

Key Concepts

Cost Function

When a company produces and sells products, like pet food, calculating costs is crucial. The cost function helps us understand how much it costs to manufacture a certain number of items. For this pet food company, the cost function is given as:\[ C = 100x + 2600 \]What this equation tells us is:

• The term \(100x\) represents the variable costs. It costs \($100\) to produce each case of pet food, so this part of the function scales with the number of cases produced.
• The term \(2600\) is the fixed cost. These are costs that remain the same no matter how many cases are produced, such as rent or salaries.

Understanding both fixed and variable costs is essential for the company to determine how profitable their business can be. It's important to know at what point the revenues will cover these costs, leading to our next topic on revenue function.

Revenue Function

The revenue function expresses how much money a company makes from selling its products. For the pet food company, each case of pet food, containing 100 boxes, is sold at \(\(120\). Thus, the revenue function is:\[ R = 120x \]What this means is:

• For each case sold, the company earns \(\)120\).
• The revenue function \(R\) directly multiplies the number of cases sold \(x\) by the revenue per case \(120\), showing their relationship is linear and directly proportional.

Revenue functions allow businesses to predict income based upon the number of units sold, aiding in strategic planning and financial forecasting. It is particularly significant when analyzing the break-even point, where revenue equals costs.

Equation Solving

To determine how many cases need to be sold to break even, we set the revenue function equal to the cost function. This gives us the following equation:\[ 120x = 100x + 2600 \]Solving this involves a few straightforward steps:

• Subtract \(100x\) from both sides to isolate the variable term on one side, leading to \(20x = 2600\).
• Now, solve for \(x\) by dividing both sides by 20, thus \(x = \frac{2600}{20}\).
• Calculating the above yields \(x = 130\). This indicates that 130 cases need to be sold to cover all costs, both fixed and variable.

This equation-solving step is vital, as it helps to determine the break-even point where the business neither gains nor loses money. Practicing these calculations can help students better understand financial planning and profitability analysis.