To find the volume of the rectangular prism, you can use the formula for the volume of a rectangular prism:
\[ \text{Volume} = \text{Length} \times \text{Width} \times \text{Height} \]
First, convert the mixed numbers into improper fractions:
- Height: \( 10 \frac{1}{2} = \frac{21}{2} \) inches
- Length: \( 6 \frac{2}{3} = \frac{20}{3} \) inches
- Width: \( 2 \frac{1}{4} = \frac{9}{4} \) inches
Now, plug these values into the volume formula:
\[ \text{Volume} = \frac{20}{3} \times \frac{9}{4} \times \frac{21}{2} \]
Calculating step by step:
- Multiply the fractions:
\[ \frac{20 \times 9 \times 21}{3 \times 4 \times 2} \]
- Calculate the numerator:
\[ 20 \times 9 = 180 \] \[ 180 \times 21 = 3780 \]
So, the numerator is \( 3780 \).
- Calculate the denominator:
\[ 3 \times 4 = 12 \] \[ 12 \times 2 = 24 \]
So, the denominator is \( 24 \).
- Now, put it all together:
\[ \text{Volume} = \frac{3780}{24} \]
- Simplify the fraction:
\[ \frac{3780 \div 12}{24 \div 12} = \frac{315}{2} \]
Thus,
\[ \text{Volume} = 157.5 \text{ cubic inches} \]
Now, we need to check if this volume can hold the item with dimensions:
- Height: \( 8 \) inches
- Length: \( 6 \) inches
- Width: \( 2 \) inches
Calculating the volume of the item:
\[ \text{Volume of item} = 6 \times 2 \times 8 = 96 \text{ cubic inches} \]
Now, we compare the volumes:
- Volume of the rectangular prism: \( 157.5 \) cubic inches
- Volume of the item: \( 96 \) cubic inches
Since \( 157.5 \text{ cubic inches} > 96 \text{ cubic inches} \), the rectangular prism is large enough to hold the item.