Chapter 12.2, Problem 27PPS

To determine

Find the diameter .

Answer to Problem 27PPS

diameter = 16 mm

Explanation of Solution

Given:

A cylinder has a surface area of $256\pi m{m}^2$ and a height of 8 mm.

Formula Used: The Lateral area L of a cylinder with radius r and height h is

  $L=2\pi rh$

The Surface area S of a cylinder with radius r and height h is

  $S=2\pi rh+2\pi r^2$

Calculation:

The given cylinder has height 8 mm and Surface area = $256\pi m{m}^2$

Surface area:

  $\begin{array}{l}S=2\pi rh+2\pi r^2=2\pi r(8)+2\pi r^2=256\pi \mathrm{....}[given]\\ \Rightarrow 2\pi (8r+r^2)=256\pi \\ \Rightarrow 8r+r^2=\frac{256\pi }{2\pi }\mathrm{........}[divide\text{ each side by }2\pi \text{]}\\ \Rightarrow 8r+r^2=128\\ \Rightarrow r^2+8r-128=\mathrm{0......}[subtract\text{ 128 from each side]}\\ \Rightarrow r^2+16r-8r-128=\mathrm{0...}\left[\begin{array}{l}\text{factor the middle term such that they multiply to -128}\\ \text{and they add to 8}\text{.}\end{array}\right]\\ \Rightarrow (r+16)(r-8)=0\\ \Rightarrow r=-16\text{ and }r=\mathrm{8......}[ab=0\iff a=0\text{ or b=0 ]}\end{array}$

Since radius cannot be negative , so, r = 8 mm.

Hence , diameter = 2(8) = 16 mm.