Respuesta :
Answer:
40,700 ft²
Step-by-step explanation:
You want the surface area of the exterior of a building in the shape of a 150 ft high square-base prism with a square pyramid on top that is 1/6 the height of the prism. The base is 60 ft square.
Pyramid slant height
The area of one face of the pyramid will be half the product of its base edge length and its slant height.
The slant height can be found from its base edge length and its height using the Pythagorean theorem.
The height of the pyramid is (1/6)(150 ft) = 25 ft. The distance from the center of the base of the pyramid to the midpoint of one side is half the side length of the square base, so is (60 ft)/2 = 30 ft. Then the slant height is ...
s = √(25² +30²) = √1525 ≈ 39.05 . . . . feet
Exterior area
The area of one side of the building is the sum of the rectangular area of the prism and the area of one face of they pyramid:
A = LW + 1/2(Ws) = W(L +1/2s)
A = (60 ft)(150 ft + 39.05/2 ft) ≈ 10,171.54 ft²
The exterior area of the building is 4 times this amount:
Exterior Area = 4(10171.54 ft²) ≈ 40686.15 ft²
You want this rounded to the nearest 100 square feet.
The exterior area of the building is about 40,700 square feet.
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Additional comment
You can reject the 18,356.47 ft² answer immediately, because it is improperly rounded. If you look at the math underlying how it was obtained, you see that it uses a formula that calculates the total area of the pyramid. The prism area isn't included, and the area of the base of the pyramid should not be included.
Each face of the building is a 60 ft by 150 ft rectangle, so has an area of 9000 ft². The answer must be greater than 4 times that, or 36000 ft².
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The surface area of the exterior of the building is approximately 83300 square feet.
To find the surface area of the exterior of the building, we need to calculate the surface area of the prism and the surface area of the pyramid separately, and then add them together.
The surface area of the prism can be found using the formula A_prism = 2lw + 2lh + 2wh, where l is the length, w is the width, and h is the height. Since the base of the prism is a square with an edge length of 60 feet, we have l = w = 60 feet. The height of the prism is given as 150 feet. Substituting these values into the formula, we get A_prism = 2(60)(60) + 2(60)(150) + 2(150)(60) = 72000 square feet.
The height of the pyramid is stated as one-sixth of the height of the prism, which is 150/6 = 25 feet. The base of the pyramid is also a square with an edge length of 60 feet. The surface area of the pyramid can be found using the formula A_pyramid = lw + 2ls, where l is the length, w is the width, and s is the slant height. The slant height can be calculated using the Pythagorean theorem, s = sqrt(l^2 + (h/2)^2) = sqrt(60^2 + (25/2)^2) ≈ 63.58 feet. Substituting the values into the formula, we get A_pyramid = (60)(60) + 2(60)(63.58) = 11229.6 square feet.
Adding the surface areas of the prism and the pyramid together, we get A_total = A_prism + A_pyramid = 72000 + 11229.6 = 83229.6 square feet.
Rounding to the nearest hundred square feet, the surface area of the exterior of the building is approximately 83300 square feet.
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