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Expert-verified Found in: Page 108 ### Calculus

Book edition 1st
Author(s) Peter Kohn, Laura Taalman
Pages 1155 pages
ISBN 9781429241861 # Find a formula for the cost $C\left(r\right)$ 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.

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

$C\left(r\right)=0.25\left(2\mathrm{\pi rh}\right)+0.50\left(2{\mathrm{\pi r}}^{2}\right)+0.1\left(2\left(2\mathrm{\pi r}\right)+h\right)$

Substitute $h=5$ in the cost function.

$30=0.25\left(2\mathrm{\pi r}\left(5\right)\right)+0.50\left(2{\mathrm{\pi r}}^{2}\right)+0.1\left(2\left(2\mathrm{\pi r}\right)+5\right)\phantom{\rule{0ex}{0ex}}30=2.5\mathrm{\pi r}+0.1{\mathrm{\pi r}}^{2}+0.4\mathrm{\pi r}+0.5\phantom{\rule{0ex}{0ex}}0=2.9\mathrm{\pi r}+0.1{\mathrm{\pi r}}^{2}+0.5-30\phantom{\rule{0ex}{0ex}}0=2.9\mathrm{\pi r}+0.1{\mathrm{\pi r}}^{2}-29.5\phantom{\rule{0ex}{0ex}}\mathrm{r}\approx 1.94$

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,

$20<|C\left(r\right)|<40\phantom{\rule{0ex}{0ex}}20-20<|C\left(r\right)-20|<40-20\phantom{\rule{0ex}{0ex}}0<|C\left(r\right)-20|<20$

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

role="math" localid="1648026622152" $|C\left(r\right)-20-10|<20-10\phantom{\rule{0ex}{0ex}}|C\left(r\right)-30|<10$

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

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

$r\in \left(1.43,2.38\right)$.

Therefore, $\left|r-1.94\right|<0.44$. ### Want to see more solutions like these? 