Chapter 1: Q. 45 (page 108)

Find a formula for the cost $C (r)$ of producing a gourmet soup can with radius $r$ and height $5$ inches, and answer the following questions:
What is the radius of a can that is $5$ inches tall and costs $30$ cents to produce?
Your manager wants you to produce $5$ -inch-tall cans that cost between $20$ and $40$ cents. Write this requirement as an absolute value inequality.
What range of radii would satisfy your manager? Write an absolute value inequality whose solution set lies inside this range of radii.

Short Answer

Expert verified

The radius of a can that is $5$ inches tall and costs $30$ cents to produce is, $1.94$ inches.
The requirement as an absolute value inequality is, $|C (r) - 30| < 10$ .
The range of radius that is satisfied is, $r \in (1.43, 2.38)$ . An absolute value inequality whose solution set lies inside this range of radii is, $|r - 1.94| < 0.44$ .

Step by step solution

Part a Step 1. Given information

Radius $= r$ .

Height $= 5$ inches.

Total cents is, $30$ .

Part a Step 2. Considering the radius r and height 5 inches, the cost function will be,

$C (r) = 0.25 (2 πrh) + 0.50 (2 {πr}^{2}) + 0.1 (2 (2 πr) + h)$

Substitute $h = 5$ in the cost function.

$30 = 0.25 (2 πr (5)) + 0.50 (2 {πr}^{2}) + 0.1 (2 (2 πr) + 5) 30 = 2.5 πr + 0.1 {πr}^{2} + 0.4 πr + 0.5 0 = 2.9 πr + 0.1 {πr}^{2} + 0.5 - 30 0 = 2.9 πr + 0.1 {πr}^{2} - 29.5 r \approx 1.94$

Hence, the radius is, $1.94$ inches.

Part b Step 1. Given information

The manager wants us to produce $5$ -inch-tall cans that cost between $20$ and $40$ cents.

Part b Step 2. Considering that the cost of cans of 5-inch tall between 20 and 40 cents, the inequality can be written as,

$20 < | C (r) | < 40 20 - 20 < | C (r) - 20 | < 40 - 20 0 < | C (r) - 20 | < 20$

Again, subtracting $10$ from each sides we get,

role="math" localid="1648026622152" $| C (r) - 20 - 10 | < 20 - 10 | C (r) - 30 | < 10$

Hence, the inequality is, $|C (r) - 30| < 10$ .

Part c Step 1. The range of radius that is satisfied is,

$r \in (1.43, 2.38)$ .

Therefore, $|r - 1.94| < 0.44$ .

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Short Answer

Step by step solution

Part a Step 1. Given information

Part a Step 2. Considering the radius r and height 5 inches, the cost function will be,

Part b Step 1. Given information

Part b Step 2. Considering that the cost of cans of 5-inch tall between 20 and 40 cents, the inequality can be written as,

Part c Step 1. The range of radius that is satisfied is,

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Discrete Mathematics

Decision Maths

Geometry

Applied Mathematics

Logic and Functions

Study anywhere. Anytime. Across all devices.