Question

What is the slant height of a square pyramid that has a surface area of 189 square feet and a side length of 7 feet? A. 14 ft B. 12 ft C. 10 ft D. 8 ft

Answer 1

C.) 10..........................

Answer 2

Answer:  The correct option is (C). 10 feet.

Step-by-step explanation:  We are given to find the slant height of a square pyramid that has a surface area of 189 square feet and a side length of 7 feet.

We know that the surface area of a square pyramid with base edge 'a' units and height 'h' units is given by

$A=a^2+2a\sqrt{\dfrac{a^2}{4}+h^2}.$

A square pyramid is shown in the attached figure.

In the given square pyramid, we have

length of the base edge, a = 7 feet,

Surface area, S.A. = 189 sq. ft.

If 'h' is the height of the pyramid, then we have

$A=a^2+2a\sqrt{\dfrac{a^2}{4}+h^2}\\\\\\\Rightarrow 189=7^2+2\times 7\sqrt{\dfrac{7^2}{4}+h^2}\\\\\\\Rightarrow 189=49+14\sqrt{\dfrac{49}{4}+h^2}\\\\\\\Rightarrow 189-49=14\sqrt{\dfrac{49}{4}+h^2}\\\\\\\Rightarrow 140=14\sqrt{\dfrac{49}{4}+h^2}\\\\\\\Rightarrow 10=\sqrt{\dfrac{49}{4}+h^2}\\\\\\\Rightarrow 100=\dfrac{49}{4}+h^2\\\\\\\Rightarrow h^2=100-\dfrac{49}{4}\\\\\\\Rightarrow h^2=\dfrac{351}{4}.$.

So, if 'l' is the slant height of the pyramid, then

$l^2=h^2+\left(\dfrac{a}{2}\right)^2\\\\\\\Rightarrow l^2=\dfrac{351}{4}+\left(\dfrac{7}{2}\right)^2\\\\\\\Rightarrow l^2=\dfrac{351}{4}+\dfrac{49}{4}\\\\\\\Rightarrow l^2=100\\\\\Rightarrow l=10.$

Thus, the slant height of the square pyramid is 10 feet.

Option (C) is correct.