Q. 46 - Calculus

Question

Find a formula for the cost $C\left(h\right)$ of producing a gourmet soup can with height $h$ and radius $2$ inches, and answer the following questions:

1. What is the height of a can that has radius $2$ inches and costs $45$ cents to produce?
2. Your manager wants you to produce $2$-inch-radius cans that cost between $40$ and $50$ cents. Write this requirement as an absolute value inequality.
3. What range of heights would satisfy your manager? Write an absolute value inequality whose solution set lies within this range of heights.

Step-by-Step Solution

Answer

1. The height of a can that has radius $2$ inches and costs $45$ cents to produce is,$9.23$ inches.
2. The requirement as an absolute value inequality is, $\left|C\left(r\right)-40\right|<10$.
3. The range of height that is to be satisfied is, $h\in \left(1.43,2.38\right)$.An absolute value inequality whose solution set lies within this range of heights is, $\left|h-1.94\right|<0.44$.

Part a Step 1 . Given information

Height $=h$.

Radius $=2$ inches.

Total cent $=45$.

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

$C\left(r\right)=0.25\left(2\mathrm{\pi rh}\right)+0.5\left(2{\mathrm{\pi r}}^2\right)+0.1\left(2\right(2\mathrm{\pi r})+h)$.

Substitute $r=5$ in the cost function.

$$\begin{gathered} 45=0.25\left(2\mathrm{\pi }\right(2\left)h\right)+0.5\left(2\mathrm{\pi }(2{)}^2\right)+0.1\left(2\right(2\mathrm{\pi }·2)+h) \\ 45=\mathrm{\pi }h+4\mathrm{\pi }+0.8\mathrm{\pi }+0.1h \\ 45=3.14 h+12.566+2.513+0.1 h \\ 45=3.24 h+15.08 \\ 3.24 h=45-15.08 \\ h=\frac{29.92}{3.24} \\ h=9.23 \end{gathered}$$

Hence, the height is, $9.23$ inches.

Part b Step 1 . Given information

The manager wants us to produce $2$-inch-radius cans that cost between $40$ and $50$ cents.

Part b Step 2 . Considering the cost of cans of 2 -inch radius between 40 and 50 cents, the inequality can be written as,

$$\begin{gathered} 40<\left|C\right(r\left)\right|<50 \\ 40-40<\left|C\right(r)-40|<50-40 \\ 0<\left|C\right(r)-40|<10 \end{gathered}$$

Hence, the inequality is, $\left|C\left(r\right)-40\right|<10$.

Part c Step 1 . The range of height that is to be satisfied is,

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

Therefore,

$\left|h-1.94\right|<0.44$.