To find 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. The base area \( B \) can be calculated as:
\[ B = \text{length} \times \text{width} \]
Given:
- Length = \( 7 \frac{1}{4} , \text{m} \) = \( \frac{29}{4} , \text{m} \) (converted to an improper fraction)
- Width = \( 10 \frac{1}{2} , \text{m} \) = \( \frac{21}{2} , \text{m} \) (converted to an improper fraction)
- Height = \( 12 \frac{2}{5} , \text{m} \) = \( \frac{62}{5} , \text{m} \) (converted to an improper fraction)
Now, calculate the base area \( B \):
\[ B = \left( \frac{29}{4} \right) \times \left( \frac{21}{2} \right) \]
Calculating the multiplication of fractions:
\[ B = \frac{29 \times 21}{4 \times 2} = \frac{609}{8} , \text{m}^2 \]
Next, calculate the volume \( V \):
\[ V = B \times h = \left( \frac{609}{8} \right) \times \left( \frac{62}{5} \right) \]
Multiplying the fractions:
\[ V = \frac{609 \times 62}{8 \times 5} = \frac{37758}{40} \]
Now, simplify \( \frac{37758}{40} \):
- Find the GCD of 37758 and 40. The GCD is 2.
- Divide both the numerator and denominator by the GCD:
\[ \frac{37758 \div 2}{40 \div 2} = \frac{18879}{20} \]
Now, convert \( \frac{18879}{20} \) to a mixed number. Divide 18879 by 20:
\[ 18879 \div 20 = 943 \quad \text{R} 19 \]
This means:
\[ \frac{18879}{20} = 943 \frac{19}{20} \]
Thus, the volume of the rectangular prism is:
\[ \boxed{943 \frac{19}{20}} , \text{m}^3 \]