Question

Multiply the edge lengths of a right rectangular prism with length 1/2 , width 1 1/2 , and height 3/4 to show that the product is the same as the volume you found by packing the prism with unit cubes with an edge length of 1/4 inch.

Cube method:
Number of cubes: 36
Volume of each cube: 1/4×1/4×1/4=1/64

Volume of the prism: v=36×1/64
=36/64
=9/16 cubic units

Answer 1

To find the volume of the right rectangular prism using the edge lengths, we can multiply the length, width, and height together.

Given:
- Length = \( \frac{1}{2} \)
- Width = \( 1 \frac{1}{2} = \frac{3}{2} \)
- Height = \( \frac{3}{4} \)

Now, we can calculate the volume \( V \) of the prism:

\[
V = \text{Length} \times \text{Width} \times \text{Height}
\]

Substituting in the values:

\[
V = \left( \frac{1}{2} \right) \times \left( \frac{3}{2} \right) \times \left( \frac{3}{4} \right)
\]

Now, we will perform the multiplication step-by-step:

1. Multiply the first two fractions:
\[
\frac{1}{2} \times \frac{3}{2} = \frac{1 \times 3}{2 \times 2} = \frac{3}{4}
\]

2. Now multiply the result by the height:
\[
\frac{3}{4} \times \frac{3}{4} = \frac{3 \times 3}{4 \times 4} = \frac{9}{16}
\]

So, the volume of the prism calculated using the edge lengths is:

\[
V = \frac{9}{16} \text{ cubic units}
\]

Now, let's compare this with the volume found using the packing method. According to the cube method:

1. The number of unit cubes (each with an edge length of \( \frac{1}{4} \)) is 36.
2. The volume of each cube is:
\[
\text{Volume of each cube} = \frac{1}{4} \times \frac{1}{4} \times \frac{1}{4} = \frac{1}{64}
\]
3. Therefore, the total volume using the cube method is:
\[
V = 36 \times \frac{1}{64} = \frac{36}{64} = \frac{9}{16} \text{ cubic units}
\]

As we can see, both methods yield the same volume:

\[
\text{Volume of the prism} = \frac{9}{16} \text{ cubic units}
\]

Thus, the product of the edge lengths confirms that the volume found through both methods is consistent.