Problem 17 - Beginning and Intermediate Algebra

Question

\(R(x)=18 x\) is the revenue function for the sale of \(x\) toasters, in dollars. The cost to manufacture \(x\) toasters, in dollars, is \(C(x)=15 x+2400\) a) Find the profit function, \(P(x)\), that describes the profit from the sale of \(x\) toasters. b) What is the profit from the sale of 800 toasters?

Step-by-Step Solution

Answer

The profit function is \(P(x) = 3x - 2400\), and the profit from the sale of 800 toasters is 0 dollars.

Step 1: Find the Profit Function (P(x))

To find the profit function, take the difference between the revenue function and the cost function. P(x) = R(x) - C(x) P(x) = 18x - (15x + 2400)

Step 2: Simplify the Profit Function

We will now simplify the profit function by combining like terms. P(x) = 18x - 15x - 2400 P(x) = 3x - 2400 Now we have found the profit function: \(P(x) = 3x - 2400\).

Step 3: Find the Profit of 800 Toasters

To find the profit from the sale of 800 toasters, substitute \(x=800\) into the profit function and calculate the result. P(800) = 3(800) - 2400 P(800) = 2400 - 2400 P(800) = 0 The profit from the sale of 800 toasters is 0 dollars.

Key Concepts

Revenue Function

The revenue function represents the total income generated from selling a certain number of products, in this case, toasters. This function is crucial for understanding how much money the sales bring in before other expenses. To write this function, we multiply the number of items sold by the price per item. In the exercise, the revenue function is given as \( R(x) = 18x \).

• The variable \( x \) signifies the number of toasters sold.
• The constant 18 indicates the selling price per toaster in dollars.

To derive such a function in general, you simply need to multiply the quantity sold (\( x \)) by the unit price. This gives a simple linear relationship between sales volume and revenue.

Cost Function

The cost function helps determine the total expenses involved in manufacturing a specified number of products. This function usually combines both fixed and variable costs. In this context, the cost function is \( C(x) = 15x + 2400 \).

• The term \( 15x \) reflects the variable cost associated with producing \( x \) toasters, meaning each toaster costs $15 to manufacture.
• The figure 2400 represents the fixed costs, such as rent and salaries, which remain constant regardless of the output level.

By understanding the components of cost, you can easily separate what will change with production volume and what will not, vital for strategic planning.

Simplifying Expressions

Simplifying expressions involves reducing them to a form that is easier to understand or use. In the problem, you are asked to simplify the profit function, \( P(x) = 18x - (15x + 2400) \).

• First, distribute the negative sign to both terms within the parentheses, resulting in \( 18x - 15x - 2400 \).
• Next, combine like terms: \( 18x - 15x \) simplifies to \( 3x \).

Thus, the profit function simplifies to \( P(x) = 3x - 2400 \). This step makes it much easier to interpret, highlighting the key relationships in the problem. A simpler expression can help you instantly identify how changes in inputs (like selling 1 more toaster) might impact the outputs (the profit).

Substitution Method

The substitution method involves replacing a variable with its corresponding numerical value to find specific results. In this exercise, after obtaining the simplified profit function, \( P(x) = 3x - 2400 \), you need to calculate the profit from selling 800 toasters.

• Substitute \( x = 800 \) into the function: \( P(800) = 3(800) - 2400 \).
• This results in \( P(800) = 2400 - 2400 \), leading to a profit of \( 0 \) dollars.

In mathematical exercises, substitution allows you to directly compute values of interest by replacing variables, making it an essential tool for analyzing different scenarios.