Chapter 11.3, Problem 41PPS

(a)

To determine

To Find: The ratio comparing the surface area to the volume then simplify. State the excluded values of the variables.

Answer to Problem 41PPS

  $\frac{5}{x};x\ne 0$

Explanation of Solution

Given: To minimize the packaging expenses a company uses packages that have the least surface area and the volume ratio.

  

The shape of the packet is cuboid

  $\begin{array}{l}length(a)=x\\ width(b)=x\\ height(c)=2x\end{array}$

Surface area of the cuboid= $2ab+2bc+2ac$

  $\begin{array}{l}=2(x)(x)+2(x)(2x)+2(x)(2x)\\ =2x^2+4x^2+4x^2\\ =10x^2\end{array}$

Volume of the cuboid = $abc$

  $\begin{array}{l}=(x)(x)(2x)\\ =2x^3\end{array}$

Then

Ratio ofsurface area to the volume is

  $=\frac{10x^2}{2x^3}=\frac{5}{x}$

For the rational expression denominator cannot be zero

For excluded vale $\begin{array}{l}2x^3=0\\ x=0\end{array}$

Thus

The excluded value is $x=0$

(b)

To determine

To Find: The ratio comparing the surface area to the volume then simplify. State the excluded values of the variables.

Answer to Problem 41PPS

  $\frac{2(a+b)}{ab};a,b\ne 0$

Explanation of Solution

Given: To minimize the packaging expenses a company uses packages that have the least surface area and the volume ratio.

  

The shape of the packet is cylinder

  $\begin{array}{l}radius(r)=a\\ height(h)=b\end{array}$

Surface area of the cylinder= $2\pi r^2+2\pi rh$

  $\begin{array}{l}=2\pi r^2+2\pi rh\\ =2\pi a^2+2\pi ab\\ =2\pi a(a+b)\end{array}$

Volume of the cylinder= $\pi r^{2}h$

  $\begin{array}{l}=\pi r^{2}h\\ =\pi a^{2}b\end{array}$

Then

Ratio ofsurface area to the volume is

  $=\frac{2\pi a(a+b)}{\pi a^{2}b}=\frac{2(a+b)}{ab}$

For the rational expression denominator cannot be zero

For excluded vale

  $\begin{array}{l}ab=0\\ a=0,b=0\end{array}$

Thus

The excluded value is $a=0,b=0$