To find the volume of the toy box, we can use the formula for the volume of a rectangular prism, which is:
\[ \text{Volume} = \text{Length} \times \text{Width} \times \text{Height} \]
Given the measurements in mixed fractions:
- Length = \(4 \frac{1}{5}\) feet
- Width = \(2 \frac{1}{3}\) feet
- Height = \(3 \frac{1}{2}\) feet
First, we convert the mixed fractions to improper fractions.
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Convert \(4 \frac{1}{5}\): \[ 4 \frac{1}{5} = \frac{(4 \times 5) + 1}{5} = \frac{20 + 1}{5} = \frac{21}{5} \]
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Convert \(2 \frac{1}{3}\): \[ 2 \frac{1}{3} = \frac{(2 \times 3) + 1}{3} = \frac{6 + 1}{3} = \frac{7}{3} \]
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Convert \(3 \frac{1}{2}\): \[ 3 \frac{1}{2} = \frac{(3 \times 2) + 1}{2} = \frac{6 + 1}{2} = \frac{7}{2} \]
Now, we can substitute these improper fractions into the volume formula:
\[ \text{Volume} = \left(\frac{21}{5}\right) \times \left(\frac{7}{3}\right) \times \left(\frac{7}{2}\right) \]
Now we multiply these fractions together:
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Multiply the numerators: \[ 21 \times 7 \times 7 = 1029 \]
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Multiply the denominators: \[ 5 \times 3 \times 2 = 30 \]
Thus, the volume is: \[ \text{Volume} = \frac{1029}{30} \]
Next, we simplify this fraction. We can perform the division to express it as a mixed fraction: \[ 1029 \div 30 = 34\quad \text{remainder: } 9 \] So, \[ \frac{1029}{30} = 34 \frac{9}{30} \]
Now, we simplify \(\frac{9}{30}\):
- The greatest common divisor of 9 and 30 is 3.
- Thus: \[ \frac{9}{30} = \frac{3}{10} \]
Bringing it all together, we have: \[ \text{Volume} = 34 \frac{3}{10} \]
Therefore, the volume of the toy box is: \[ \boxed{34 \frac{3}{10}} \text{ cubic feet} \]