Chapter 2, Problem 98RE

Solution

a.

To find: An expression of the surface area in terms of radius $x$ .volume of Cylindrical can is $1-L$

The expression for surface area $S=2\pi x^2+\frac{2000}{x}$ .

Given information:

The volume is equal to $1-L$ and radius $r=x.$

Formula used:

Volume $V=\pi r^{2}h$

Surface area $S=2\pi r^2+2\pi rh$

Calculations:

The volume:

  $\begin{array}{l}1000c{m}^3=\pi x^{2}h\mathrm{Sin}ce1L=1000c{m}^{3}andr=x\\ heighth=\frac{1000}{x^2\pi }\end{array}$

Then surface area:

  $\begin{array}{c}S=2\pi r^2+2\pi rh\\ =2\pi x^2+2\pi x\times \frac{1000}{x^2\pi }\\ =2\pi x^2+\frac{2000}{x}\end{array}$

Hence the expression for surface area $S=2\pi x^2+\frac{2000}{x}$ .

b.

To find : the radius and height if the surface area is $900c{m}^{2.}$ , $S=2\pi x^2+\frac{2000}{x}$ .

The height $h=60.20cm$ if the radius $x=2.3cm$ and the radius $r=10.64cm$ then height $h=2.3cm$

Given information:

The volume is equal to $1-L$ and radius $r=x.$ , $S=2\pi x^2+\frac{2000}{x}$ .

Formula used:

Volume $V=\pi r^{2}h$

Surface area $S=2\pi x^2+\frac{2000}{x}$

  $h=\frac{1000}{x^2\pi }$

Calculations:

Substitute the values into equation of surface area.

  $\begin{array}{c}S=2\pi x^2+\frac{2000}{x}\\ 900=2\pi x^2+\frac{2000}{x}\\ 2\pi x^3-900x=-2000\\ x\left(3.14x^2-450\right)=-1000\\ x=10.64,-12.96,2.30\end{array}$

Since radius is the positive value so if $r=x=10.64cm$ .

Then height $h=\frac{1000}{x^2\pi }$ substitute the values then,

  $\begin{array}{c}h=\frac{1000}{3.14\times {10.64}^2}\\ =2.8cm\end{array}$

if $r=x=2.3cm$ .

Then height $h=\frac{1000}{x^2\pi }$ substitute the values then,

  $\begin{array}{c}h=\frac{1000}{3.14\times {2.3}^2}\\ =60.20cm\end{array}$

So height $h=60.20cm$ if the radius $x=2.3cm$ and the radius $r=10.64cm$ then height $h=2.3cm$ .