Question

select the correct answer.

a silo shown in the diagram is to be built out of aluminum without any gaps or overlaps.

what is the amount of aluminum needed to the nearest square meter?

\\(2,487\text{ m}^2\\)

\\(2,600\text{ m}^2\\)

\\(2,713\text{ m}^2\\)

\\(2,374\text{ m}^2\\)

Explanation:

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<reasoning>

Identify the geometric components and dimensions

\[

$$\begin{aligned} &d = 12\text{ m} \implies r = 6\text{ m} \\ &h_{\text{total}} = 60\text{ m} \\ &h_{\text{cylinder}} = h_{\text{total}} - r = 60 - 6 = 54\text{ m} \end{aligned}$$

\]

Calculate the surface area of the components

\[

$$\begin{aligned} &A_{\text{dome}} = 2\pi r^2 = 2\pi (6)^2 = 72\pi\text{ m}^2 \\ &A_{\text{lateral cylinder}} = 2\pi r h_{\text{cylinder}} = 2\pi (6)(54) = 648\text{ m}^2 \\ &A_{\text{base}} = \pi r^2 = \pi (6)^2 = 36\pi\text{ m}^2 \end{aligned}$$

\]

Calculate the total surface area

\[

$$\begin{aligned} &A_{\text{total}} = A_{\text{dome}} + A_{\text{lateral cylinder}} + A_{\text{base}} \\ &A_{\text{total}} = 72\pi + 648\pi + 36\pi = 756\pi\text{ m}^2 \\ &A_{\text{total}} \approx 756 \times 3.14159265 \approx 2375.04\text{ m}^2 \approx 2374\text{ m}^2 \text{ (using } \pi \approx 3.14\text{)} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(2,487\text{ m}^2\)</mcq-option>
<mcq-option>(B) \(2,600\text{ m}^2\)</mcq-option>
<mcq-option>(C) \(2,713\text{ m}^2\)</mcq-option>
<mcq-correct>(D) \(2,374\text{ m}^2\)</mcq-correct>
</answer>

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"Surface Area of Composite Solids"
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}
</post_analysis>

Answer:

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<reasoning>

Identify the geometric components and dimensions

\[

$$\begin{aligned} &d = 12\text{ m} \implies r = 6\text{ m} \\ &h_{\text{total}} = 60\text{ m} \\ &h_{\text{cylinder}} = h_{\text{total}} - r = 60 - 6 = 54\text{ m} \end{aligned}$$

\]

Calculate the surface area of the components

\[

$$\begin{aligned} &A_{\text{dome}} = 2\pi r^2 = 2\pi (6)^2 = 72\pi\text{ m}^2 \\ &A_{\text{lateral cylinder}} = 2\pi r h_{\text{cylinder}} = 2\pi (6)(54) = 648\text{ m}^2 \\ &A_{\text{base}} = \pi r^2 = \pi (6)^2 = 36\pi\text{ m}^2 \end{aligned}$$

\]

Calculate the total surface area

\[

$$\begin{aligned} &A_{\text{total}} = A_{\text{dome}} + A_{\text{lateral cylinder}} + A_{\text{base}} \\ &A_{\text{total}} = 72\pi + 648\pi + 36\pi = 756\pi\text{ m}^2 \\ &A_{\text{total}} \approx 756 \times 3.14159265 \approx 2375.04\text{ m}^2 \approx 2374\text{ m}^2 \text{ (using } \pi \approx 3.14\text{)} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(2,487\text{ m}^2\)</mcq-option>
<mcq-option>(B) \(2,600\text{ m}^2\)</mcq-option>
<mcq-option>(C) \(2,713\text{ m}^2\)</mcq-option>
<mcq-correct>(D) \(2,374\text{ m}^2\)</mcq-correct>
</answer>

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