Problem 88 - College Algebra and Calculus: An Applied Approach

Question

The revenue \(R\) for selling \(x\) units of a product is \(R=25.95 x .\) The cost \(C\) of producing \(x\) units is $$ C=13.95 x+125,000 $$ In order to obtain a profit, the revenue must be greater than the cost. For what values of \(x\) will this product return a profit?

Step-by-Step Solution

Answer

The product will return a profit for values of \( x \) greater than \( 10416.67 \).

Step 1: Set up the Profit Inequality

First, we need to express the notion of profit in mathematical terms. Profit is defined as Revenue subtracted by Cost. So, the condition for a profit is when \( R > C \). Substitute the given equations into this inequality: \( 25.95x > 13.95x + 125,000 \).

Step 2: Simplify the Inequality

To solve for \( x \), gather like terms: Subtract \( 13.95x \) from both sides to get \( 12x > 125,000 \).

Step 3: Solve the Inequality

Now divide both sides by \( 12 \) to isolate \( x \): \( x > 10416.67 \).

Key Concepts

Profit Calculation

When it comes to running a business, understanding how to calculate profit is critical. Profit is the money that remains once you have subtracted all the costs from your revenues. In mathematical terms, we express this idea with the formula:

• Profit = Revenue - Cost.

In this exercise, we are tasked with determining when a product will be profitable based on given cost and revenue functions. This means we need to find out under what conditions our revenue exceeds our costs. By setting up and solving an inequality, we can find the number of units needed to make a profit.
Understanding profit calculation is not just useful for giant corporations, but it's also essential for small businesses and startups looking to manage their budget and resources effectively.

Revenue and Cost Functions

To determine profit, we must understand the revenue and cost functions, which are crucial components of any business equation. These two functions tell us how much money is coming in and going out.

• Revenue Function (R): This function shows how much money is made from selling a particular number of products. In our example, it is given by: \[ R = 25.95x \]where \( x \) is the number of units sold and 25.95 is the price per unit.
• Cost Function (C): This function reflects the total expenses of producing a number of products, expressed as:\[ C = 13.95x + 125,000 \]The first term 13.95x represents the variable cost per unit, while 125,000 is the fixed cost.

Grasping these functions is key to strategizing pricing, budgeting, and financial forecasting. By comparing these functions, businesses can make informed decisions to ensure profitability.

Solving Inequalities

Inequalities, like equations, are a fundamental part of algebra, and solving them is essential for practical applications such as profit analysis. Unlike equations that assert equality, inequalities show that one side is either greater than or less than the other. Our task here is to find when the revenue exceeds the cost, expressed as \( R > C \).
The solution involves a few steps:

• First, substitute the expressions for revenue and cost into the inequality: \[ 25.95x > 13.95x + 125,000 \]
• Simplify by gathering like terms, subtracting \( 13.95x \) from both sides: \[ 12x > 125,000 \]
• Divide both sides of the inequality by 12 to find the number of units needed for profit:\[ x > 10416.67 \]

This tells us that more than 10,416.67 units must be sold to achieve profitability. While solving inequalities may seem daunting at first, understanding and accurately applying these steps can help in answering critical business questions.