**Problem 2: Manufacturing Cost Analysis for Silly Soaker**The manufacturer of the water toy "Silly Soaker" quotes a variable cost of $5.25 per unit and fixed costs of $7,000.**a. Create a function to represent the average cost per unit to manufacture the Silly Soaker.**To find the average cost per unit, we start by defining the total cost function, $ C(x) $, which is the sum of the fixed costs and the variable costs. It is given by:\[ C(x) = 7000 + 5.25x \]The average cost per unit, $ A(x) $, is then:\[ A(x) = \frac{C(x)}{x} = \frac{7000 + 5.25x}{x} \]So, the function for the average cost per unit is:\[ A(x) = \frac{7000}{x} + 5.25 \]**b. Use that model to determine the average cost per unit for a level of production of $ x = 5,000 $ units.**Substitute $ x = 5000 $ into the average cost function:\[ A(5000) = \frac{7000}{5000} + 5.25 = 1.4 + 5.25 = 6.65 \]Therefore, the average cost per unit for producing 5,000 units is \$6.65.**c. What is the horizontal asymptote of this function, and what does it represent?**The horizontal asymptote of the function $ A(x) = \frac{7000}{x} + 5.25 $ is determined by evaluating the function as $ x $ approaches infinity. As $ x $ increases, $ \frac{7000}{x} $ approaches 0. Therefore, the horizontal asymptote is:\[ y = 5.25 \]This horizontal asymptote represents the minimum average cost per unit, or the variable cost per unit, once the production level is sufficiently large. In essence, as production increases, the fixed costs become negligible per unit, leaving the average cost per unit to approach the variable cost of \$5.25.

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2. The manufacturer of the water toy "Silly Soaker" quotes a variable cost of $5.25 per unit and fixed costs of $7,000. a. Create a function to represent the average cost per unit to manufacture the Silly Soaker. b. Use that model to determine the average cost per unit for a level of production of x = 5,000 units. c. What is the horizontal asymptote of this function, and what does it represent?

2. The manufacturer of the water toy "Silly Soaker" quotes a variable cost of $5.25 per unit and fixed costs of $7,000. a. Create a function to represent the average cost per unit to manufacture the Silly Soaker. b. Use that model to determine the average cost per unit for a level of production of x = 5,000 units. c. What is the horizontal asymptote of this function, and what does it represent?

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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Question

**Problem 2: Manufacturing Cost Analysis for Silly Soaker**

The manufacturer of the water toy "Silly Soaker" quotes a variable cost of $5.25 per unit and fixed costs of $7,000.

**a. Create a function to represent the average cost per unit to manufacture the Silly Soaker.**

To find the average cost per unit, we start by defining the total cost function, $ C(x) $, which is the sum of the fixed costs and the variable costs. It is given by:

\[ C(x) = 7000 + 5.25x \]

The average cost per unit, $ A(x) $, is then:

\[ A(x) = \frac{C(x)}{x} = \frac{7000 + 5.25x}{x} \]

So, the function for the average cost per unit is:

\[ A(x) = \frac{7000}{x} + 5.25 \]

**b. Use that model to determine the average cost per unit for a level of production of $ x = 5,000 $ units.**

Substitute $ x = 5000 $ into the average cost function:

\[ A(5000) = \frac{7000}{5000} + 5.25 = 1.4 + 5.25 = 6.65 \]

Therefore, the average cost per unit for producing 5,000 units is \$6.65.

**c. What is the horizontal asymptote of this function, and what does it represent?**

The horizontal asymptote of the function $ A(x) = \frac{7000}{x} + 5.25 $ is determined by evaluating the function as $ x $ approaches infinity. As $ x $ increases, $ \frac{7000}{x} $ approaches 0. Therefore, the horizontal asymptote is:

\[ y = 5.25 \]

This horizontal asymptote represents the minimum average cost per unit, or the variable cost per unit, once the production level is sufficiently large. In essence, as production increases, the fixed costs become negligible per unit, leaving the average cost per unit to approach the variable cost of \$5.25.

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