Problem 95 - Algebra for College Students

Question

Solve each problem. Average cost. Bobby spent \(\$ 150\) on \(x\) pieces of clothing for her child. a) Find a rational function \(C(x)\) that gives the average cost in dollars for an item of clothing. b) Find \(C(5), C(10),\) and \(C(30)\)

Step-by-Step Solution

Answer

The rational function is \ C(x) = \frac{150}{x} \. \ C(5) = 30 \; \ C(10) = 15 \; \ C(30) = 5 \.

Step 1 - Determine the Rational Function

To find the average cost per item, divide the total cost by the number of items. The rational function for average cost, where total cost is \$150\ and the number of items is \(x\), is: \[ C(x) = \frac{150}{x} \]

Step 2 - Calculate C(5)

Plug \(x = 5\) into the rational function. \[ C(5) = \frac{150}{5} = 30 \] So, the average cost for 5 items is \$30\.

Step 3 - Calculate C(10)

Plug \(x = 10\) into the rational function. \[ C(10) = \frac{150}{10} = 15 \] So, the average cost for 10 items is \$15\.

Step 4 - Calculate C(30)

Plug \(x = 30\) into the rational function. \[ C(30) = \frac{150}{30} = 5 \] So, the average cost for 30 items is \$5\.

Key Concepts

rational function

A rational function is a fraction where both the numerator and the denominator are polynomials. In the context of the problem, the numerator is the total cost (which is a constant value of 150), and the denominator is the variable number of items, denoted as \(x\). Hence, the rational function can be written as: \[ C(x) = \frac{150}{x} \]
This function shows the average cost per item as we change the number of items \(x\). The key property of rational functions is that they can represent relationships where a quantity is distributed across a varying number of items. As \(x\) increases, the average cost per item decreases.

cost per item

The cost per item is essentially the average cost that Bobby pays for each piece of clothing. The rational function derived earlier gives this cost.
Let's understand it with some specific values:

• When \(C(5) = 30\), it means the cost per item when buying 5 pieces of clothing is \(30 each.
• When \(C(10) = 15\), it means the cost per item when buying 10 pieces of clothing is \)15 each.
• When \(C(30) = 5\), it means the cost per item when buying 30 pieces of clothing is $5 each.

Notice how the cost per item decreases as the number of items increases. This is a typical inverse relationship represented by rational functions.

algebraic calculations

Algebraic calculations are key to solving problems involving rational functions. Let's break down the steps used in solving the exercise:

• First, we derive the rational function based on the given information. Here, it was \( C(x) = \frac{150}{x} \).
• Next, we plug in specific values for \(x\) to calculate the average cost.

Let's see how it's done in each case:

• To find \( C(5) \), substitute 5 into the function: \[ C(5) = \frac{150}{5} = 30\]
• For \( C(10) \), substitute 10 into the function: \[ C(10) = \frac{150}{10} = 15\]
• And for \( C(30) \), substitute 30 into the function: \[ C(30) = \frac{150}{30} = 5\]

By understanding these steps, you can solve similar problems independently!