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?

$2,487\text{ m}^2$

$2,600\text{ m}^2$

$2,713\text{ m}^2$

$2,374\text{ m}^2$

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?

$2,487\text{ m}^2$

$2,600\text{ m}^2$

$2,713\text{ m}^2$

$2,374\text{ m}^2$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [], "new_concepts": [ "Surface Area of Composite Solids", "Surface Area of Cylinders", "Surface Area of Hemispheres" ], "current_concepts": [ "Surface Area of Composite Solids", "Surface Area of Cylinders", "Surface Area of Hemispheres" ] } ## 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 $$ \begin{aligned} &A_{\text{total}} = A_{\text{dome}} + A_{\text{lateral cylinder}} + A_{\text{base}} \ &A_{\text{total}} = 72\pi + 648\pi + 36\pi = 756\pi\text{ m}^2 \ &A_{\text{total}} \approx 756 \times 3.14159265 \approx 2375.04\text{ m}^2 \approx 2374\text{ m}^2 \text{ (using } \pi \approx 3.14\text{)} \end{aligned} $$

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

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

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

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

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Geometry", "Surface Area of Composite Solids" ] }

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QUESTION IMAGE

Question

Explanation:

Identify the geometric components and dimensions

Calculate the surface area of the components

Calculate the total surface area

Answer: