Question
If the rectangular prism shown were dilated by a scale factor of 12 . What would the volume of the dilated prism be? (1 point) Responses 120 cm2 120 cm2 15 cm2 15 cm2 60 cm2 60 cm2 30 cm2 30 cm2. L 5cm W 6cm H 4cm. Pick one answer even if wrong
Answers
Answered by
GPT-4o mini
To determine the volume of the dilated rectangular prism, we first need to calculate the original volume of the prism.
The volume \( V \) of a rectangular prism is given by the formula:
\[
V = L \times W \times H
\]
where \( L \), \( W \), and \( H \) are the length, width, and height, respectively.
Given:
- Length \( L = 5 \) cm
- Width \( W = 6 \) cm
- Height \( H = 4 \) cm
Calculating the original volume:
\[
V = 5 \times 6 \times 4 = 120 \text{ cm}^3
\]
When a solid is dilated by a scale factor \( k \), the new volume \( V' \) can be calculated using the formula:
\[
V' = k^3 \times V
\]
In this case, the scale factor \( k = 12 \).
Calculating the new volume:
\[
V' = 12^3 \times 120
\]
\[
12^3 = 1728
\]
\[
V' = 1728 \times 120 = 207360 \text{ cm}^3
\]
None of the options provided seem to correlate with this calculation. However, if we purely look at the options given without the calculations, the original volume of the prism is \( 120 \text{ cm}^3 \), which could be suggested as an intermediate step rather than the dilated volume.
But since the response options are limited to much smaller numbers (like \( 120 \text{ cm}^2 \), \( 60 \text{ cm}^2 \), etc.), it seems like a miscommunication or miscalculation may have occurred when considering or presenting the options.
If I had to pick from the given choices, I would suggest possibly picking:
**120 cm²**
However, understand that this doesn't reflect the volume of the dilated man. The correct dilated volume as per the calculation is \( 207360 \text{ cm}^3 \), assuming all parameters and scale factors were communicated accurately.
The volume \( V \) of a rectangular prism is given by the formula:
\[
V = L \times W \times H
\]
where \( L \), \( W \), and \( H \) are the length, width, and height, respectively.
Given:
- Length \( L = 5 \) cm
- Width \( W = 6 \) cm
- Height \( H = 4 \) cm
Calculating the original volume:
\[
V = 5 \times 6 \times 4 = 120 \text{ cm}^3
\]
When a solid is dilated by a scale factor \( k \), the new volume \( V' \) can be calculated using the formula:
\[
V' = k^3 \times V
\]
In this case, the scale factor \( k = 12 \).
Calculating the new volume:
\[
V' = 12^3 \times 120
\]
\[
12^3 = 1728
\]
\[
V' = 1728 \times 120 = 207360 \text{ cm}^3
\]
None of the options provided seem to correlate with this calculation. However, if we purely look at the options given without the calculations, the original volume of the prism is \( 120 \text{ cm}^3 \), which could be suggested as an intermediate step rather than the dilated volume.
But since the response options are limited to much smaller numbers (like \( 120 \text{ cm}^2 \), \( 60 \text{ cm}^2 \), etc.), it seems like a miscommunication or miscalculation may have occurred when considering or presenting the options.
If I had to pick from the given choices, I would suggest possibly picking:
**120 cm²**
However, understand that this doesn't reflect the volume of the dilated man. The correct dilated volume as per the calculation is \( 207360 \text{ cm}^3 \), assuming all parameters and scale factors were communicated accurately.
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