Chapter 5.3, Problem 31PS

Solution

(a)

To find an equation that gives the length.

  $\text{l=}\frac{{\text{100ft}}^{\text{3}}}{{\text{πr}}^{\text{2}}}$

Given information: ${\text{V=πr}}^{\text{2}}\text{h}$

Formula used: Cylinder volume is ${\text{V=πr}}^{\text{2}}\text{l}$

Calculation:

Cylinder volume is ${\text{V=πr}}^{\text{2}}\text{h}$ , where $\text{r}$ is the radius of the each circular end and $\text{h}$ is the distance between the two circular ends ("height" when the cylinder is standing on one end). $\text{l}$ corresponds to $\text{h}$ in this case.

Solve the volume formula for $\text{l}$ , with ${\text{V=100ft}}^{\text{3}}$

  $\begin{array}{c}\text{l=}\frac{\text{V}}{{\text{πr}}^{\text{2}}}\\ \text{l=}\frac{{\text{100ft}}^{\text{3}}}{{\text{πr}}^{\text{2}}}\end{array}$

Hence length is $\text{l=}\frac{{\text{100ft}}^{\text{3}}}{{\text{πr}}^{\text{2}}}$

(b)

To find a function that gives the surface area.

  $\text{S=2}\pi {\text{r}}^2\text{+}\frac{{\text{200ft}}^{\text{3}}}{\text{r}}$

Substitute for $\text{l}$ , the formula we got from sub-part (a).

  $\begin{array}{l}\text{d=1000to1300}\\ \text{50}\\ {\text{4}}^{\text{o}}\text{C}\end{array}$

Hence, ${\text{S=2πr}}^{\text{2}}\text{+}\frac{{\text{200ft}}^{\text{3}}}{\text{r}}$

(c)

Find the dimensions.

  $\text{r}\approx \text{2}\text{.52ft, l}\approx \text{5}\text{.03ft}$

  

  $\text{S=2}\pi {\text{r}}^2\text{+}\frac{{\text{200ft}}^{\text{3}}}{\text{r}}$

Use a graph to get the minimum point.

To find the minimum on interval $\text{2<x<3}$ :

The least amount of plastic used is when $\text{r}\approx \text{2}\text{.52ft}$ for a total of $\text{119}{\text{.27ft}}^{\text{2}}$ of plastic used.

Then for $\text{l}$ :

  $\text{l=}\frac{{\text{100ft}}^{\text{3}}}{{\text{πr}}^{\text{2}}}=\frac{{\text{100ft}}^{\text{3}}}{\text{π}{\left(2.52\right)}^{\text{2}}}\approx 5.03\text{ft}$

Hence , $\text{r}\approx \text{2}\text{.52ft, l}\approx \text{5}\text{.03ft}$