Chapter 2, Problem 99RE

Solution

a.

To find: An expression for the surface e area as a function of $x$ and base volume of rectangular tank with square base and no top is $1000f{t}^3$ of water.

The expression for surface area $S=\frac{4000}{x}+x^2$ .

Given information:

The volume is equal to $1000f{t}^3$ of water.

And base length is $x$ .

Formula used:

Volume $V=basearea\times height$.

Surface area $S=4\times areaoftheside+basearea$

Calculations:

The volume:

  $\begin{array}{l}1000f{t}^3=x^{2}h\mathrm{Sin}ce1L=1000f{t}^{3}andbasearea\\ heighth=\frac{1000}{x^2}\end{array}$

Then surface area:

  $\begin{array}{c}S=4x\times \frac{1000}{x^2}+x^2\\ =\frac{4000}{x}+x^2\end{array}$

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

Hence the expression for surface area $S=\frac{4000}{x}+x^2$ .

b.

To find : the length,width and height if the out side surface area is $600f{t}^{2.}$ , $S=x^2+\frac{4000}{x}$ .

If length $$

  $20ft,$ width $20ft$ then height will be $2.5ft.$ and if length $$

  $7.32ft,$ width $7.3ft$ then height is about $18.66ft.$

Given information:

The volume is equal to $1000f{t}^3$ of water.

And base length is $x$ .

Formula used:

Volume $V=basearea\times height$

Surface area $S=4\times areaoftheside+basearea$

Calculations:

Surface area :

  $\begin{array}{c}S=x^2+\frac{4000}{x}\\ 600=x^2+\frac{4000}{x}\\ \Rightarrow x^3-600x+4000=0\\ x=20,-27,7.32\end{array}$

Since length, height,width are positive values so take positive values.

For the length

  $\begin{array}{c}x=20,\\ h=\frac{1000}{{20}^2}\\ =2.5ft\end{array}$

For the length

  $\begin{array}{c}x=7.32,\\ h=\frac{1000}{{7.32}^2}\\ =18.66ft\end{array}$

So if length $$ $20ft,$ width $20ft$ then height will be $2.5ft.$ and if length $$ $7.32ft,$ width $7.3ft$ then height is about $18.66ft.$