Questions: The dimensions of a rectangular shipping crate are 2.5 ft., 2.5 ft., and 4 ft. If there were a similar shipping crate with a scale factor of 3, what would be the volume of the new shipping crate? (1 point) 225 ft^3 75 ft^3 25 ft^3 675 ft^3

The dimensions of a rectangular shipping crate are 2.5 ft., 2.5 ft., and 4 ft. If there were a similar shipping crate with a scale factor of 3, what would be the volume of the new shipping crate? (1 point) 225 ft^3 75 ft^3 25 ft^3 675 ft^3
Transcript text: The dimensions of a rectangular shipping crate are 2.5 ft ., 2.5 ft ., and 4 ft . If there were a similar shipping crate with a scale factor of 3 , what would be the volume of the new shipping crate? (1 point) $225 \mathrm{ft}^{3}{ }^{3}$ $75 \mathrm{ft}^{3}$ $25 \mathrm{ft}^{3}{ }^{3}$ $675 \mathrm{ft}^{3}{ }^{3}$
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Solution

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Solution Steps

Step 1: Identify the dimensions of the original crate

The original crate has dimensions of \( 2.5 \, \text{ft} \), \( 2.5 \, \text{ft} \), and \( 4 \, \text{ft} \).

Step 2: Calculate the volume of the original crate

The volume \( V \) of a rectangular prism is given by: \[ V = \text{length} \times \text{width} \times \text{height} \] Substituting the given dimensions: \[ V = 2.5 \times 2.5 \times 4 = 25 \, \text{ft}^3 \]

Step 3: Apply the scale factor to find the volume of the new crate

The scale factor is \( 3 \). Since volume scales with the cube of the scale factor, the new volume \( V' \) is: \[ V' = V \times (\text{scale factor})^3 \] Substituting the values: \[ V' = 25 \times 3^3 = 25 \times 27 = 675 \, \text{ft}^3 \]

Final Answer

\(\boxed{675 \, \text{ft}^3}\)

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