Chapter 12.7, Problem 33SR

Solution

To find: the lateral and total surface area.

The lateral area is $\text{374}{\text{.4ft}}^2.$ .

Total surface area is $\text{499}{\text{.2ft}}^2$

Given:

Height = $10.4$ feet, length $12$ feet, prism height $3\frac{1}{3}$ feet.

Formulas used:

The lateral area $\text{L}$ of a triangular prism is $\text{L=Ph}$

The surface area $S$ of a triangular prism is $S=L+2B$

Calculation:

Consider an equilateral triangular prism whose base height is $10.4\text{ ft}\text{.}$ , base length is $\text{12 ft}\text{.}$ and the height of the prism is $\text{3}\frac{1}{3}\text{ ft}\text{.}$ .

The objective is to find the lateral and surface area of the triangular prism.

First convert the mixed number into decimal number.

  $\begin{array}{l}\text{3}\frac{1}{3}=\frac{3.3+1}{3}\\ \text{ = }\frac{9+1}{3}\\ \text{ =}\frac{10}{3}\\ \text{ =3}\text{.33}\end{array}$

The lateral area $\text{L}$ of a triangular prism is $\text{L=Ph}$

Here, $\text{P}$ is the perimeter of the base of the triangle and $h$ is the height the triangular prism.

The surface area $S$ of a triangular prism is $S=L+2B$

Here, $L$ is the lateral area and $B$ is the area of the triangular base.

The base of the prism is a triangle whose sides are $12\text{ft},\text{ 12ft}$ and $\text{12ft}$

Find the lateral area of the triangular prism as follows:

The perimeter $P$ of the base of the prism is,

  $P$ =Sum of the three sides of the triangle.

Therefore,

  $\begin{array}{l}P=12+12+12\\ \text{ =36}\end{array}$

Substitute $\text{P=36}$ and $\text{h=10}\text{.4}$ in $\text{L=P}\text{.h}$

  $\begin{array}{l}\text{L=P}\text{.h}\\ \text{ =(36)(10}\text{.4)}\\ \text{ =374}\text{.4}\end{array}$

Hence, the lateral area of the triangular prism is $\text{374}{\text{.4ft}}^2.$

Find the surface area of the triangle prism as follows:

The area $B$ of the base is $B=\frac{1}{2}.b.a$

Here, $b$ is the base and $a$ is the height of the triangle.

Substitute $b=12$ , $a=10.4$ in $B=\frac{1}{2}.b.a$

  $\begin{array}{l}B=\frac{1}{2}.b.a\\ \text{ =}\frac{1}{2}.(12).(10.4)\\ \text{ =(6)(10}\text{.4)}\\ \text{ =62}\text{.4}\end{array}$

Therefore, $\text{B=62}\text{.4}$

Substitute $\text{B=62}\text{.4}$ and

  $\text{L=374}\text{.4}$ in $\text{S=L+2B}$

  $\begin{array}{l}\text{S=L+2B}\\ \text{ =374}\text{.4+2(62}\text{.4)}\\ \text{ =374}\text{.4+124}\text{.8}\\ \text{ =499}\text{.2}\end{array}$

Hence, the surface area of the triangular prism is $\text{499}{\text{.2ft}}^2$ .