Question

Use the image to answer the question.

An illustration shows a 3 D rectangular cuboid segmented into 5 rows by 2 columns of blocks. The length is start fraction 1 over 2 end fraction, width is start fraction 1 over 4 end fraction, and height is start fraction 5 over 4 end fraction.

Multiply the edge lengths of a right rectangular prism with length 1/2
m, width 1/4
m, and height 5/4
m to show that the product is the same as the volume found by packing the prism with 10 unit cubes with an edge length of 1/4
m. What is the volume of the prism?

(1 point)
cubic meters
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Answer 1

To find the volume of the right rectangular prism with the given dimensions, we use the formula for the volume of a rectangular prism:

\[
\text{Volume} = \text{length} \times \text{width} \times \text{height}
\]

Given:

- Length \( = \frac{1}{2} \) m
- Width \( = \frac{1}{4} \) m
- Height \( = \frac{5}{4} \) m

Now substituting these values into the volume formula:

\[
\text{Volume} = \left(\frac{1}{2}\right) \times \left(\frac{1}{4}\right) \times \left(\frac{5}{4}\right)
\]

Calculating step by step:

1. First, calculate \( \frac{1}{2} \times \frac{1}{4} \):

\[
\frac{1}{2} \times \frac{1}{4} = \frac{1 \times 1}{2 \times 4} = \frac{1}{8}
\]

2. Now, multiply \( \frac{1}{8} \) by \( \frac{5}{4} \):

\[
\frac{1}{8} \times \frac{5}{4} = \frac{1 \times 5}{8 \times 4} = \frac{5}{32}
\]

Thus, the volume of the prism is:

\[
\text{Volume} = \frac{5}{32} \text{ cubic meters}
\]

To check if this volume matches that of packing the prism with unit cubes, we calculate the volume of one unit cube with an edge length of \( \frac{1}{4} \) m:

\[
\text{Volume of one unit cube} = \left(\frac{1}{4}\right) \times \left(\frac{1}{4}\right) \times \left(\frac{1}{4}\right) = \frac{1}{64} \text{ cubic meters}
\]

Since we have 10 of these unit cubes:

\[
\text{Total volume of 10 unit cubes} = 10 \times \frac{1}{64} = \frac{10}{64} = \frac{5}{32} \text{ cubic meters}
\]

Since both methods yield the same result:

\[
\frac{5}{32} \text{ cubic meters}
\]

Thus, the volume of the prism is \( \frac{5}{32} \) cubic meters.