Question

a square pyramid has a height of 25 feet and each side of the base has a length of 15 ft. if a model of the square pyramid is scaled down by a factor of $\frac{1}{5}$, what is the surface area of the model? 161.28 ft.$^{2}$ 64.05 ft.$^{2}$ 256.20 ft.$^{2}$ 40.32 ft.$^{2}$

Explanation:

Step1: Find the slant height of the original pyramid

First, find the distance from the center of the base to the mid - point of a side of the base. For a square base of side length $s = 15$ ft, this distance $a=\frac{s}{2}=\frac{15}{2}=7.5$ ft. Using the Pythagorean theorem $l=\sqrt{h^{2}+a^{2}}$, where $h = 25$ ft. So $l=\sqrt{25^{2}+7.5^{2}}=\sqrt{625 + 56.25}=\sqrt{681.25}\approx26.1$ ft.

Step2: Calculate the surface area of the original pyramid

The surface area of a square pyramid $S = s^{2}+2sl$, where $s = 15$ ft and $l\approx26.1$ ft. So $S=15^{2}+2\times15\times26.1=225 + 783=1008$ ft².

Step3: Apply the scale - factor

If the scale factor is $k=\frac{1}{5}$, the surface area of the scaled - down model $S_{model}=S\times k^{2}$. Since $S = 1008$ ft² and $k=\frac{1}{5}$, then $S_{model}=1008\times(\frac{1}{5})^{2}=1008\times\frac{1}{25}=40.32$ ft².

Answer:

40.32 ft²