Question

type the correct answer in the box. use numerals instead of words. a new building is formed by a square prism with a square pyramid on top. the base has an edge - length of 60 feet, and the height of the prism is 150 feet. the height of the pyramid is one - sixth the height of the prism. what is the surface area of the exterior of the building rounded to the nearest hundred square feet? the surface area of the exterior of the building is approximately square feet.

Explanation:

Step1: Calculate surface - area of prism

The surface - area of a square prism (excluding the top face that is in contact with the pyramid) is $S_{prism}=4\times(edge\times height)+(edge)^2$. Here, $edge = 60$ feet and $height = 150$ feet. So, $S_{prism}=4\times(60\times150)+60^{2}=4\times9000 + 3600=36000+3600 = 39600$ square feet.

Step2: Calculate slant - height of pyramid

The height of the pyramid $h_p=\frac{1}{6}\times150 = 25$ feet. The base edge of the pyramid $a = 60$ feet. The slant - height $l$ of the square pyramid is calculated using the Pythagorean theorem: $l=\sqrt{(\frac{a}{2})^{2}+h_p^{2}}=\sqrt{(\frac{60}{2})^{2}+25^{2}}=\sqrt{900 + 625}=\sqrt{1525}\approx39.05$ feet.

Step3: Calculate surface - area of pyramid (excluding the base)

The surface - area of the square pyramid (excluding the base) is $S_{pyramid}=4\times(\frac{1}{2}\times edge\times l)$. Substituting $edge = 60$ feet and $l\approx39.05$ feet, we get $S_{pyramid}=4\times(\frac{1}{2}\times60\times39.05)=4\times1171.5 = 4686$ square feet.

Step4: Calculate total surface - area of the building

The total surface - area of the building $S=S_{prism}+S_{pyramid}=39600+4686=44286\approx44300$ square feet.

Answer:

44300