Chapter 12.1, Problem 31WE

To determine

To find: the total area of the solid.

Answer to Problem 31WE

The total area of the solid is $198{\text{cm}}^2$ .

Explanation of Solution

Given information:

The length of the rectangle solid is twice the width and the height is three times the width and the volume is $162{\text{cm}}^3$ .

Calculation: let the width of the rectangle is $x$ ,

Thus, the length of the rectangle solid is $2x$ and the height of rectangle solid is $3x$ .

Since, the volume of rectangle is $162{\text{cm}}^3$ ,

  $\begin{array}{l}v=bh\\ 162=\left(2x·x\right)\left(3x\right)\text{ [substitute]}\\ 162=6x^3\text{ [multiply]}\\ x^3=27\text{ [divide]}\\ x=3\text{ [cube root]}\end{array}$

Thus, the total surface area is.

  $\begin{array}{l}\text{T}\text{.A}\text{.}=\text{L}\text{.A}\text{.+2B}\\ \text{T}\text{.A}\text{.}=\left(x+2x+x+2x\right)\left(3x\right)\text{+2}\left(x·2x\right)\text{ [simplify]}\\ \text{T}\text{.A}\text{.}=18x^2+4x^2\text{ [multiply]}\\ \text{T}\text{.A}\text{.}=22x^2\text{ [add]}\\ \text{T}\text{.A}\text{.}=22{\left(3\right)}^2\text{ [substitute]}\\ \text{T}\text{.A}\text{.}=198{\text{cm}}^2\text{ [multiply]}\end{array}$

Thus, the total area is $198{\text{cm}}^2$ .