Question

question 9 (5 points)
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the lower base of the frustum shown in the figure above is a square 8 ft on each side. the upper base is a square 4 ft on each side. if the altitude is 6 ft, find the slant height.

Explanation:

Step1: Calculate the difference in half - side lengths

The half - side length of the lower base \(a_1=\frac{8}{2} = 4\) ft, and the half - side length of the upper base \(a_2=\frac{4}{2}=2\) ft. The difference in half - side lengths \(d=a_1 - a_2=4 - 2=2\) ft.

Step2: Use the Pythagorean theorem to find the slant height

We know that the slant height \(l\) of the frustum (a right - angled triangle is formed with the altitude \(h = 6\) ft and the difference in half - side lengths \(d\) as the legs). By the Pythagorean theorem \(l=\sqrt{h^{2}+d^{2}}\). Substitute \(h = 6\) ft and \(d = 2\) ft into the formula: \(l=\sqrt{6^{2}+2^{2}}=\sqrt{36 + 4}=\sqrt{40}=2\sqrt{10}\approx6.32\) ft.

Answer:

The slant height is \(2\sqrt{10}\approx6.32\) ft.