To determine how many \( \frac{1}{3} \)-inch cubes fit inside the given right rectangular prism, we first need to calculate the volume of the prism and the volume of one cube.
Step 1: Calculate the volume of the rectangular prism.
The dimensions of the prism are:
- Length = \( 1 \frac{1}{3} \) inches = \( \frac{4}{3} \) inches
- Width = \( 1 \) inch
- Height = \( \frac{2}{3} \) inch
The volume \( V \) of the rectangular prism is given by the formula:
\[ V = \text{Length} \times \text{Width} \times \text{Height} \]
Substituting the values:
\[ V = \left( \frac{4}{3} \right) \times 1 \times \left( \frac{2}{3} \right) \]
Calculating this:
\[ V = \frac{4}{3} \times \frac{2}{3} = \frac{8}{9} \text{ cubic inches} \]
Step 2: Calculate the volume of one \( \frac{1}{3} \)-inch cube.
The volume \( V \) of one cube with side length \( \frac{1}{3} \) inches is given by:
\[ V = \text{side}^3 = \left( \frac{1}{3} \right)^3 = \frac{1}{27} \text{ cubic inches} \]
Step 3: Determine how many cubes fit in the prism.
To find the number of \( \frac{1}{3} \)-inch cubes that can fit in the prism, divide the volume of the prism by the volume of one cube:
\[ \text{Number of cubes} = \frac{\text{Volume of prism}}{\text{Volume of one cube}} = \frac{\frac{8}{9}}{\frac{1}{27}} \]
To divide by a fraction, we multiply by its reciprocal:
\[ \frac{8}{9} \times \frac{27}{1} = \frac{8 \times 27}{9} = \frac{216}{9} = 24 \]
So, the number of \( \frac{1}{3} \)-inch cubes that would fit inside the right rectangular prism is 24 cubes.