Chapter 12.5, Problem 8PPS

To determine

To find: the lateral and surface area of the given prism.

Answer to Problem 8PPS

The lateral area of the triangular prism is $540{\text{yd}}^2$ and the surface area of the triangular prism is $1152{\text{yd}}^2$ .

Explanation of Solution

Given:

  

Formula used:

The lateral area L of a triangular prism is, $L=P\cdot h$ .

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

Calculation:

The given figure is a triangular prism.

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

The lateral area L of a triangular prism is, $L=P\cdot h$ .

Here, 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.

Find the lateral area of the triangular prism as follows:

The base of the prism is a triangle whose sides are 51 yd, 20 yd and 37 yd.

The perimeter P of the base of the prism is, $P=\left(l_1+l_2+l_3\right)$ .

Here, $l_1,l_2,l_3$ are sides of the triangle.

  $\begin{array}{c}P=\left(l_1+l_2+l_3\right)\\ =51+20+37\\ =108\end{array}$

Substitute P = 108 and h = 5 in $L=P\cdot h$

  $\begin{array}{c}L=P\cdot h\\ =\left(108\right)\left(5\right)\\ =540\end{array}$

Therefore, the lateral area of the triangular prism is $540{\text{yd}}^2$ .

Find the surface area of the triangular prism is,

Find the area B of the base of the prism is, $B=\frac{1}{2}\cdot b\cdot a$ .

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

  $\begin{array}{c}B=\frac{1}{2}\cdot b\cdot a\\ =\frac{1}{\cancel{2}}\cdot \left(51\right)\cdot \left(\cancel{1}{\cancel{2}}^6\right)\\ =51\cdot 6\\ =306\end{array}$

Substitute $B=306$ and $L=540$ in $S=L+2B$

  $\begin{array}{c}S=L+2B\\ =540+2\left(306\right)\\ =540+612\\ =1152\end{array}$

Therefore, the surface area of the triangular prism is $1152{\text{yd}}^2$ .

Conclusion:

Therefore, the lateral area of the triangular prism is $540{\text{yd}}^2$ and the surface area of the triangular prism is $1152{\text{yd}}^2$ .