Question

question 14 (5 points)
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what is the volume of a frustum of a rectangular pyramid if one base has measurements 6 ft. × 8 ft., the other base has measurements 14 ft. × 18 ft., and the height is 3 ft.?
a) 560.30 ft.³
b) 252.8 ft.³
c) 409.98 ft.³
d) 120.98 ft.³

Explanation:

Step1: Calculate the areas of the two bases

The area of a rectangle is \(A = l\times w\).
For the first base with \(l = 6\) ft and \(w = 8\) ft, \(A_1=6\times8 = 48\) \(ft^{2}\).
For the second base with \(l = 14\) ft and \(w = 18\) ft, \(A_2=14\times18=252\) \(ft^{2}\).

Step2: Use the frustum - of - pyramid volume formula

The volume formula for the frustum of a pyramid is \(V=\frac{h}{3}(A_1 + A_2+\sqrt{A_1A_2})\).
We know \(h = 3\) ft, \(A_1 = 48\) \(ft^{2}\), and \(A_2=252\) \(ft^{2}\).
First, calculate \(\sqrt{A_1A_2}=\sqrt{48\times252}=\sqrt{12096}=109.98\) (approx).
Then \(A_1 + A_2+\sqrt{A_1A_2}=48 + 252+109.98=409.98\).
Now, \(V=\frac{3}{3}\times409.98\).

Answer:

C. \(409.98\) \(ft^{3}\)