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
[tex]A=a^2+2a\sqrt{\dfrac{a^2}{4}+h^2}.[/tex]
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
[tex]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}.[/tex].
So, if 'l' is the slant height of the pyramid, then
[tex]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.[/tex]
Thus, the slant height of the square pyramid is 10 feet.
Option (C) is correct.
