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Chapter 9: Problem 69

To print t-shirts, there is a \(\$ 100\) set-up fee, plus a \(\$ 12\) charge per t-shirt. Let \(x\) represent the number of t-shirts printed and \(f(x)\) represent the total charge. (a) Write a linear function that models this situation. (b) Find \(f(125)\). Interpret the answer in the context of this problem. (c) Find the value of \(x\) if \(f(x)=1000\). Express this situation using function notation, and interpret it in the context of this problem.

Short Answer

Expert verified
The linear function is \( f(x) = 12x + 100 \). For 125 t-shirts, the total charge is \$1600. If the total charge is \$1000, then 75 t-shirts were printed.

Step by step solution

01

- Define the Linear Function

Identify the linear function based on the given information. The total charge consists of a set-up fee and a variable cost per t-shirt. The formula will have the form: \[ f(x) = mx + b \]where:- \(m\) is the charge per t-shirt, which is \$12,- \(b\) is the set-up fee, which is \$100.
02

- Write the Linear Function

Substitute the values for \(m\) and \(b\) into the formula from Step 1:\[ f(x) = 12x + 100 \]
03

- Find \(f(125)\)

Substitute \(x = 125\) into the linear function to find the total charge for printing 125 t-shirts:\[ f(125) = 12(125) + 100 \]Calculate the value:\[ f(125) = 1500 + 100 = 1600 \]Hence, the total charge for printing 125 t-shirts is \$1600.
04

- Interpret \(f(125)\)

The value \(f(125) = 1600\) means that the total cost of printing 125 t-shirts is \$1600.
05

- Find the value of \(x\) when \(f(x) = 1000\)

Set up the equation based on the given function and solve for \(x\):\[ 1000 = 12x + 100 \]First, isolate \(12x\) by subtracting 100 from both sides:\[ 900 = 12x \]Then, divide both sides by 12:\[ x = 75 \]
06

- Interpret the value of \(x = 75\)

The value \(x = 75\) means that if the total charge is \$1000, then 75 t-shirts were printed.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

linear equations
A linear equation is an equation that describes a straight line. In the context of our t-shirt printing problem, our linear equation is used to calculate the total cost based on the number of t-shirts printed. The general form of a linear equation is:

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Most popular questions from this chapter

Let \(f(x)=x^{2}\) and \(g(x)=2 x-1 .\) Match each expression in Column \(I\) with the description of how to evaluate it in Column II. II A. Square \(5 .\) Take the result and square it. B. Double 5 and subtract \(1 .\) Take the result and square it. C. Double 5 and subtract 1 . Take the result, double it, and subtract 1 . D. Square \(5 .\) Take the result, double it, and subtract 1 . I $$ (f \circ f)(5) $$

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