Question

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a dog box is built to fit in the back of a pick-up truck. it is used to transport sled dogs and supplies to a race. a dog box that holds 3 dogs is 117 cm long, 97 cm wide, and 61 cm tall. each dog compartment is 38 cm long, 97 cm wide, and 46 cm tall.
a) what is the volume of each dog compartment?
b) what is the volume of the dog box that is not used to hold dogs? how did you find out?

Explanation:

<pre_analysis>
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</pre_analysis>

<reasoning>

Calculate the volume of each dog compartment

\[

$$\begin{aligned} l &= 38\text{ cm} \\ w &= 97\text{ cm} \\ h &= 46\text{ cm} \\ V_{\text{compartment}} &= l \times w \times h \\ V_{\text{compartment}} &= 38 \times 97 \times 46 \\ V_{\text{compartment}} &= 169{,}556\text{ cm}^3 \end{aligned}$$

\]

Calculate the total volume of the dog box

\[

$$\begin{aligned} L &= 117\text{ cm} \\ W &= 97\text{ cm} \\ H &= 61\text{ cm} \\ V_{\text{total}} &= L \times W \times H \\ V_{\text{total}} &= 117 \times 97 \times 61 \\ V_{\text{total}} &= 692{,}289\text{ cm}^3 \end{aligned}$$

\]

Calculate the unused volume of the dog box

\[

$$\begin{aligned} V_{\text{used}} &= 3 \times V_{\text{compartment}} \\ V_{\text{used}} &= 3 \times 169{,}556 = 508{,}668\text{ cm}^3 \\ V_{\text{unused}} &= V_{\text{total}} - V_{\text{used}} \\ V_{\text{unused}} &= 692{,}289 - 508{,}668 \\ V_{\text{unused}} &= 183{,}621\text{ cm}^3 \end{aligned}$$

\]
</reasoning>

<answer>

Question a

The volume of each dog compartment is \(169{,}556\text{ cm}^3\).

Question b

The volume of the dog box that is not used to hold dogs is \(183{,}621\text{ cm}^3\).

This was found by calculating the total volume of the dog box (\(117\text{ cm} \times 97\text{ cm} \times 61\text{ cm} = 692{,}289\text{ cm}^3\)) and subtracting the combined volume of the three dog compartments (\(3 \times 169{,}556\text{ cm}^3 = 508{,}668\text{ cm}^3\)).
</answer>

<post_analysis>
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"subject": "Mathematics",
"question_type": "Multi-part",
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"Mathematics",
"Geometry",
"Volume of Rectangular Prism"
]
}
</post_analysis>

Answer:

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"step_cot",
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"needs_drawing": false,
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"new_concepts": [
"Volume of Rectangular Prism",
"Volume Difference"
],
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}
</pre_analysis>

<reasoning>

Calculate the volume of each dog compartment

\[

$$\begin{aligned} l &= 38\text{ cm} \\ w &= 97\text{ cm} \\ h &= 46\text{ cm} \\ V_{\text{compartment}} &= l \times w \times h \\ V_{\text{compartment}} &= 38 \times 97 \times 46 \\ V_{\text{compartment}} &= 169{,}556\text{ cm}^3 \end{aligned}$$

\]

Calculate the total volume of the dog box

\[

$$\begin{aligned} L &= 117\text{ cm} \\ W &= 97\text{ cm} \\ H &= 61\text{ cm} \\ V_{\text{total}} &= L \times W \times H \\ V_{\text{total}} &= 117 \times 97 \times 61 \\ V_{\text{total}} &= 692{,}289\text{ cm}^3 \end{aligned}$$

\]

Calculate the unused volume of the dog box

\[

$$\begin{aligned} V_{\text{used}} &= 3 \times V_{\text{compartment}} \\ V_{\text{used}} &= 3 \times 169{,}556 = 508{,}668\text{ cm}^3 \\ V_{\text{unused}} &= V_{\text{total}} - V_{\text{used}} \\ V_{\text{unused}} &= 692{,}289 - 508{,}668 \\ V_{\text{unused}} &= 183{,}621\text{ cm}^3 \end{aligned}$$

\]
</reasoning>

<answer>

Question a

The volume of each dog compartment is \(169{,}556\text{ cm}^3\).

Question b

The volume of the dog box that is not used to hold dogs is \(183{,}621\text{ cm}^3\).

This was found by calculating the total volume of the dog box (\(117\text{ cm} \times 97\text{ cm} \times 61\text{ cm} = 692{,}289\text{ cm}^3\)) and subtracting the combined volume of the three dog compartments (\(3 \times 169{,}556\text{ cm}^3 = 508{,}668\text{ cm}^3\)).
</answer>

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