Question

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your answer is incorrect.

• (a): your answer is incorrect.

answer the following questions.
write your answers in terms of π.
give exact answers (not decimal approximations).
(a) the square pyramid and the cone below both have a height of 11 mm.
the pyramid has a base length of 2√π mm, and the base of the cone has
a radius of 2 mm. a plane parallel to the bases crosses both solids at
4 mm from the top. the resulting cross sections (shaded) have the same
area.
for each solid, the top portion (which has the highlighted cross section as
its base) is similar to the entire solid.
use this fact to find the areas of the cross sections.

Explanation:

Step1: Find the scale factor for the pyramid

The height of the top - portion of the pyramid is \(h_1 = 4\) mm, and the height of the entire pyramid is \(H=11\) mm.
The scale factor \(k\) for similar solids (the ratio of the heights of the similar parts) is \(k=\frac{h_1}{H}=\frac{4}{11}\).
The base area of the entire pyramid \(A_{base - pyramid}=(2\sqrt{\pi})^2 = 4\pi\) \(mm^2\).
For similar solids, if the scale factor of the linear dimensions is \(k\), the ratio of the areas of similar cross - sections is \(k^{2}\). Let the area of the cross - section of the pyramid be \(A_{pyramid}\). Then \(A_{pyramid}=A_{base - pyramid}\times k^{2}\).
Substitute \(A_{base - pyramid}=4\pi\) and \(k = \frac{4}{11}\) into the formula: \(A_{pyramid}=4\pi\times(\frac{4}{11})^2\).

$$A_{pyramid}=4\pi\times\frac{16}{121}=\frac{64\pi}{121}$$

Step2: Find the scale factor for the cone

The height of the top - portion of the cone is \(h_1 = 4\) mm, and the height of the entire cone is \(H = 11\) mm. The scale factor \(k=\frac{h_1}{H}=\frac{4}{11}\).
The base area of the entire cone \(A_{base - cone}=\pi r^{2}\), where \(r = 2\) mm. So \(A_{base - cone}=\pi\times(2)^2=4\pi\) \(mm^2\).
For similar solids, if the scale factor of the linear dimensions is \(k\), the ratio of the areas of similar cross - sections is \(k^{2}\). Let the area of the cross - section of the cone be \(A_{cone}\). Then \(A_{cone}=A_{base - cone}\times k^{2}\).
Substitute \(A_{base - cone}=4\pi\) and \(k=\frac{4}{11}\) into the formula: \(A_{cone}=4\pi\times(\frac{4}{11})^2\).

$$A_{cone}=4\pi\times\frac{16}{121}=\frac{64\pi}{121}$$

Answer:

The area of the cross - section of the square pyramid is \(\frac{64\pi}{121}\) \(mm^{2}\), and the area of the cross - section of the cone is \(\frac{64\pi}{121}\) \(mm^{2}\).