Chapter 8.3, Problem 4P

To determine

To find:The total surface area of the cylinder.

Answer to Problem 4P

  $4\pi x^2+20\pi x+24\pi$

Explanation of Solution

Given information:The radius of the cylinder is $x+2$ and the height is $x+4$ .

Calculation:The surface area of a cylinder with radius r and height h is $2\pi r^2+2\pi rh$ .

Then, the surface area with radius $x+2$ and height $x+4$

  $\begin{array}{l}=2\pi r^2+2\pi rh\\ $ =2\pi {(x+2)}^2+2\pi (x+2)(x+4)$$ =2\pi (x^2+4x+4)+2\pi (x^2+4x+2x+8)[\because {(a+b)}^2=a^2+2ab+b^2]$=2\pi (x^2+4x+4)+2\pi (x^2+6x+8)\\ =2\pi x^2+8\pi x+8\pi +2\pi x^2+12\pi x+16\pi [\text{distributive law]}\\ \text{=}(2\pi x^2+2\pi x^2)+(8\pi x+12\pi x)+(8\pi +16\pi )[\text{combining like terms]}\\ =4\pi x^2+20\pi x+24\pi \end{array}$