To find the volume of a rectangular prism, you can use the formula:
\[ \text{Volume} = \text{Length} \times \text{Width} \times \text{Height} \]
First, let's convert the mixed numbers into improper fractions to facilitate the calculation:
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Height: \[ 10 \frac{1}{2} = \frac{21}{2} \text{ inches} \]
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Length: \[ 6 \frac{2}{3} = \frac{20}{3} \text{ inches} \]
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Width: \[ 2 \frac{1}{4} = \frac{9}{4} \text{ inches} \]
Now substituting these values into the volume formula:
\[ \text{Volume} = \left(\frac{20}{3}\right) \times \left(\frac{9}{4}\right) \times \left(\frac{21}{2}\right) \]
Now, multiply the numerators and the denominators:
\[ \text{Numerator: } 20 \times 9 \times 21 = 3780 \] \[ \text{Denominator: } 3 \times 4 \times 2 = 24 \]
Thus, the volume is:
\[ \text{Volume} = \frac{3780}{24} \text{ in}^3 \]
Now, we simplify the fraction:
Dividing both the numerator and the denominator by 12:
\[ \frac{3780 \div 12}{24 \div 12} = \frac{315}{2} = 157.5 \text{ in}^3 \]
Now we can compare this volume to the item that needs to fit inside. The dimensions of the item are:
- Height: 8 in.
- Length: 6 in.
- Width: 2 in.
The volume of the item can be calculated as:
\[ \text{Volume of the item} = 8 \times 6 \times 2 = 96 \text{ in}^3 \]
Since the volume of the prism \(157.5 , \text{in}^3\) is greater than the volume of the item \(96 , \text{in}^3\), the item will fit inside the prism.
Thus, the answer is:
\[ 157 \frac{1}{2} , \text{in}^3 \text{ and yes, the item should fit inside the prism.} \]