Question

The length of a rectangular solid is twice the width, and the height is three times the width. If the volume is 162 cm³, find the total area of the solid.

Short answer

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

Step-by-step solution

Step 1. Given information.

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

Step 2. Consider the variable.

Let the width of the rectangle is x,

Then, the length of the rectangle solid is2x and the height of rectangle solid is3x .

Step 3. Determine the total area.

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

$\begin{array}{c}v=bh\\ 162=(2x\times x)\left(3x\right)\text{[substitute]}\\ 162=6x^3\text{[multiply]}\\ x^3=27\text{[divide]}\\ x=3\text{[cuberoot]}\end{array}$

Then, the total surface area is:

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

Substitute the value of x.

$\begin{array}{c}\text{T.A.}=22{\left(3\right)}^2\text{[substitute]}\\ =198{\text{cm}}^2\text{[multiply]}\end{array}$

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