Question

determine the volume of the shaded region. when appropriate, use the \\(\pi\\) key on your calculator.

the volume of the shaded region is

Explanation:

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"concepts_used": [
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<reasoning>

Identify the geometric solids

Using the Volume of Prism knowledge point
\[

$$\begin{aligned} &V_{\text{outer}} = w \cdot h \cdot l \\ &V_{\text{inner}} = w_{\text{inner}} \cdot h_{\text{inner}} \cdot l \end{aligned}$$

\]

Calculate the outer prism volume

Using the Volume of Prism knowledge point
\[

$$\begin{aligned} &w = 8\text{ ft},\quad h = 8\text{ ft},\quad l = 18\text{ ft} \\ &V_{\text{outer}} = 8 \cdot 8 \cdot 18 = 1152\text{ ft}^3 \end{aligned}$$

\]

Calculate the inner prism volume

Using the Volume of Prism knowledge point
\[

$$\begin{aligned} &w_{\text{inner}} = 4\text{ ft},\quad h_{\text{inner}} = 4\text{ ft},\quad l = 18\text{ ft} \\ &V_{\text{inner}} = 4 \cdot 4 \cdot 18 = 288\text{ ft}^3 \end{aligned}$$

\]

Subtract to find the shaded volume

Subtract the inner hollow volume from the outer volume to find the remaining shaded region:
\[

$$\begin{aligned} &V_{\text{shaded}} = V_{\text{outer}} - V_{\text{inner}} \\ &V_{\text{shaded}} = 1152 - 288 = 864\text{ ft}^3 \end{aligned}$$

\]
</reasoning>

<answer>
Determine the volume of the shaded region. When appropriate, use the \(\pi\) key on your calculator.

The volume of the shaded region is <blank>864</blank> <blank>\(\text{ft}^3\)</blank>
</answer>

<post_analysis>
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"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Geometry",
"Composite Volume"
]
}
</post_analysis>

Answer:

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<reasoning>

Identify the geometric solids

Using the Volume of Prism knowledge point
\[

$$\begin{aligned} &V_{\text{outer}} = w \cdot h \cdot l \\ &V_{\text{inner}} = w_{\text{inner}} \cdot h_{\text{inner}} \cdot l \end{aligned}$$

\]

Calculate the outer prism volume

Using the Volume of Prism knowledge point
\[

$$\begin{aligned} &w = 8\text{ ft},\quad h = 8\text{ ft},\quad l = 18\text{ ft} \\ &V_{\text{outer}} = 8 \cdot 8 \cdot 18 = 1152\text{ ft}^3 \end{aligned}$$

\]

Calculate the inner prism volume

Using the Volume of Prism knowledge point
\[

$$\begin{aligned} &w_{\text{inner}} = 4\text{ ft},\quad h_{\text{inner}} = 4\text{ ft},\quad l = 18\text{ ft} \\ &V_{\text{inner}} = 4 \cdot 4 \cdot 18 = 288\text{ ft}^3 \end{aligned}$$

\]

Subtract to find the shaded volume

Subtract the inner hollow volume from the outer volume to find the remaining shaded region:
\[

$$\begin{aligned} &V_{\text{shaded}} = V_{\text{outer}} - V_{\text{inner}} \\ &V_{\text{shaded}} = 1152 - 288 = 864\text{ ft}^3 \end{aligned}$$

\]
</reasoning>

<answer>
Determine the volume of the shaded region. When appropriate, use the \(\pi\) key on your calculator.

The volume of the shaded region is <blank>864</blank> <blank>\(\text{ft}^3\)</blank>
</answer>

<post_analysis>
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