To calculate the volume \( V \) of a rectangular prism using the formula \( V = B \times h \), where \( B \) is the area of the base and \( h \) is the height, we first need to determine the area of the base \( B \). For a rectangular prism, \( B \) is given by the formula:
\[ B = \text{length} \times \text{width} \]
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Convert the dimensions from mixed numbers to improper fractions:
- Length: \( 7 \frac{1}{4} = \frac{29}{4} \) (since \( 7 \times 4 + 1 = 29 \))
- Width: \( 10 \frac{1}{2} = \frac{21}{2} \) (since \( 10 \times 2 + 1 = 21 \))
- Height: \( 12 \frac{2}{5} = \frac{62}{5} \) (since \( 12 \times 5 + 2 = 62 \))
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Calculate the area of the base \( B \):
\[ B = \frac{29}{4} \times \frac{21}{2} \]
To multiply the fractions, multiply the numerators and denominators:
\[ B = \frac{29 \times 21}{4 \times 2} = \frac{609}{8} \]
- Now calculate the volume \( V \):
\[ V = B \times h = \frac{609}{8} \times \frac{62}{5} \]
Again, multiply the fractions:
\[ V = \frac{609 \times 62}{8 \times 5} = \frac{37758}{40} \]
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Simplify the fraction \( \frac{37758}{40} \):
Calculate \( 37758 \div 2 = 18879 \) and \( 40 \div 2 = 20 \)
Thus,
\[ V = \frac{18879}{20} \]
- Convert \( \frac{18879}{20} \) to a mixed number:
To do this, divide 18879 by 20:
\[ 18879 \div 20 = 943.95 \implies 943 \text{ whole parts} \]
The remainder:
\[ 18879 - (943 \times 20) = 18879 - 18860 = 19 \]
Thus, the volume in mixed number form is:
\[ 943 \frac{19}{20} \]
So, the volume of the rectangular prism is:
\[ \boxed{943 \frac{19}{20}} , m^3 \]