Question

the radius of the large sphere is double the radius of the small sphere.
how many times is the volume of the large sphere than the small sphere?
2
4
6
8

Explanation:

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"concepts_used": [
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"new_concepts": [
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"Similar Solids Ratio"
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"Volume Scaling of Solids",
"Similar Solids Ratio"
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<reasoning>

Define the given radii

Using the Volume of a Sphere knowledge point
\[

$$\begin{aligned} &r_{\text{small}} = 3\\ &r_{\text{large}} = 6 \end{aligned}$$

\]

Calculate the volume of each sphere

Using the Volume of a Sphere knowledge point
\[

$$\begin{aligned} &V_{\text{small}} = \frac{4}{3}\pi (3)^3 = \frac{4}{3}\pi (27) = 36\pi\\ &V_{\text{large}} = \frac{4}{3}\pi (6)^3 = \frac{4}{3}\pi (216) = 288\pi \end{aligned}$$

\]

Find the ratio of the volumes

Using the Volume of a Sphere knowledge point
\[

$$\begin{aligned} &\text{Ratio} = \frac{V_{\text{large}}}{V_{\text{small}}} = \frac{288\pi}{36\pi} = 8 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>2</mcq-option>
<mcq-option>4</mcq-option>
<mcq-option>6</mcq-option>
<mcq-correct>8</mcq-correct>
</answer>

<post_analysis>
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"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
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"Volume of a Sphere"
]
}
</post_analysis>

Answer:

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"Volume Scaling of Solids",
"Similar Solids Ratio"
]
}
</pre_analysis>

<reasoning>

Define the given radii

Using the Volume of a Sphere knowledge point
\[

$$\begin{aligned} &r_{\text{small}} = 3\\ &r_{\text{large}} = 6 \end{aligned}$$

\]

Calculate the volume of each sphere

Using the Volume of a Sphere knowledge point
\[

$$\begin{aligned} &V_{\text{small}} = \frac{4}{3}\pi (3)^3 = \frac{4}{3}\pi (27) = 36\pi\\ &V_{\text{large}} = \frac{4}{3}\pi (6)^3 = \frac{4}{3}\pi (216) = 288\pi \end{aligned}$$

\]

Find the ratio of the volumes

Using the Volume of a Sphere knowledge point
\[

$$\begin{aligned} &\text{Ratio} = \frac{V_{\text{large}}}{V_{\text{small}}} = \frac{288\pi}{36\pi} = 8 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>2</mcq-option>
<mcq-option>4</mcq-option>
<mcq-option>6</mcq-option>
<mcq-correct>8</mcq-correct>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Geometry",
"Volume of a Sphere"
]
}
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