Chapter 12.7, Problem 29STP

To determine

To find: the slant height of the pyramid

Answer to Problem 29STP

The slant height of the square pyramid is 10 m.

Explanation of Solution

Given:

A square pyramid has a base with sides measuring 7 meters. The surface area of the pyramid is 189 square meters.

1. 8 m
2. 10 m
3. 14 m
4. 16 m

Formula used:

The lateral area Lof a square pyramidis $L=\frac{1}{2}Pl$ .

Here, P is the perimeter of the base and l is the slant height the square pyramid.

The surface area S of a Square pyramid is S = L + B.

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

Calculation:

Consider a square pyramid whose sides length of the base is 7 m and the surface area of the pyramid is $189m^2$ .

The objective is to find the slant height of the square pyramid.

Find the lateral area of the square pyramid as follows:

The side of the square is 7 m.

The perimeter P of the square is, P = 4 (side).

Substitute side = 7 in P = 4(side).

Thus,

  $\begin{array}{l}P=4.7\\ =28\end{array}$

Assume the slant height of the square pyramid is l.

Substitute $P=28$ in $L=\frac{1}{2}Pl$ .

  $\begin{array}{l}L=\frac{1}{2}Pl\\ =\frac{1}{\cancel{2}}\cdot 28^{14}\cdot l\\ =14l\end{array}$

Hence, the lateral area L, of a square pyramid is 14l.

Find the surface area of the square pyramid as follows:

The area B of the base is $B={\left(side\right)}^2$

Substitute side = 7 in $B={\left(side\right)}^2$

  $\begin{array}{l}B=7^2\\ =49\end{array}$

Substitute B=49, L = 14 l and S = 189 in S = L + B.

  $\begin{array}{l}S=L+B\\ 189=14l+49\\ 189-49=14l+49-49\\ 140=14l\\ 14l=140\\ l=\frac{140}{14}\\ =10\end{array}$

Hence, the slant height of the square pyramid is 10 m.

Conclusion:

Hence, the option B is the correct one.