Question

determine (a) the volume and (b) the surface area of the three-dimensional figure. use the \\(\pi\\) button on your calculator.

a) the volume of the cylinder is
(round to the nearest hundredth as needed.)

b) the surface area of the cylinder is
(round to the nearest hundredth as needed.)

Explanation:

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"concepts_used": [],
"new_concepts": [
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"Surface Area of Cylinder"
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}
</pre_analysis>

<reasoning>

Identify cylinder dimensions

\[

$$\begin{aligned} r &= 6\text{ in.} \\ h &= 12\text{ in.} \end{aligned}$$

\]

Calculate the volume

\[

$$\begin{aligned} V &= \pi r^2 h \\ V &= \pi (6)^2 (12) = 432\pi \\ V &\approx 1357.1680... \approx 1357.17\text{ in}^3 \end{aligned}$$

\]

Calculate the surface area

\[

$$\begin{aligned} A &= 2\pi r h + 2\pi r^2 \\ A &= 2\pi (6)(12) + 2\pi (6)^2 = 144\pi + 72\pi = 216\pi \\ A &\approx 678.5840... \approx 678.58\text{ in}^2 \end{aligned}$$

\]
</reasoning>

<answer>

Question a

The volume of the cylinder is <blank>1357.17</blank> cubic inches.

Question b

The surface area of the cylinder is <blank>678.58</blank> square inches.
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multi-part",
"knowledge_point": [
"Mathematics",
"Geometry",
"Volume and Surface Area of Cylinders"
]
}
</post_analysis>

Answer:

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"step_cot",
"explain_then_answer",
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],
"needs_drawing": false,
"concepts_used": [],
"new_concepts": [
"Volume of Cylinder",
"Surface Area of Cylinder"
],
"current_concepts": [
"Volume of Cylinder",
"Surface Area of Cylinder"
]
}
</pre_analysis>

<reasoning>

Identify cylinder dimensions

\[

$$\begin{aligned} r &= 6\text{ in.} \\ h &= 12\text{ in.} \end{aligned}$$

\]

Calculate the volume

\[

$$\begin{aligned} V &= \pi r^2 h \\ V &= \pi (6)^2 (12) = 432\pi \\ V &\approx 1357.1680... \approx 1357.17\text{ in}^3 \end{aligned}$$

\]

Calculate the surface area

\[

$$\begin{aligned} A &= 2\pi r h + 2\pi r^2 \\ A &= 2\pi (6)(12) + 2\pi (6)^2 = 144\pi + 72\pi = 216\pi \\ A &\approx 678.5840... \approx 678.58\text{ in}^2 \end{aligned}$$

\]
</reasoning>

<answer>

Question a

The volume of the cylinder is <blank>1357.17</blank> cubic inches.

Question b

The surface area of the cylinder is <blank>678.58</blank> square inches.
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multi-part",
"knowledge_point": [
"Mathematics",
"Geometry",
"Volume and Surface Area of Cylinders"
]
}
</post_analysis>