To find the volume \( V \) of a rectangular prism using the formula \( V = B \times h \), where \( B \) is the area of the base (length × width) and \( h \) is the height, we will first convert the mixed numbers into improper fractions, then calculate the area of the base and finally the volume.
Let's break down the dimensions:
-
Length: \( 7 \frac{1}{4} = \frac{29}{4} \) meters (since \( 7 \times 4 + 1 = 29\))
-
Width: \( 10 \frac{1}{2} = \frac{21}{2} \) meters (since \( 10 \times 2 + 1 = 21\))
-
Height: \( 12 \frac{2}{5} = \frac{62}{5} \) meters (since \( 12 \times 5 + 2 = 62\))
Next, calculate the area of the base \( B \):
\[ B = \text{Length} \times \text{Width} = \frac{29}{4} \times \frac{21}{2} \]
Calculating the fraction multiplication:
\[ B = \frac{29 \times 21}{4 \times 2} = \frac{609}{8} \quad \text{(since } 29 \times 21 = 609 \text{ and } 4 \times 2 = 8\text{)} \]
Now, we will calculate the volume \( V \):
\[ V = B \times h = \frac{609}{8} \times \frac{62}{5} \]
Calculating this step-by-step:
- Multiply the numerators:
\[ 609 \times 62 = 37758 \]
- Multiply the denominators:
\[ 8 \times 5 = 40 \]
- Thus, the volume \( V \) is:
\[ V = \frac{37758}{40} \]
Next, we need to simplify this fraction. First, we can perform the division:
\[ 37758 \div 40 = 943.95 \]
To convert \( \frac{37758}{40} \) to a mixed number, divide:
- Find the whole number part:
\[ 943\text{ whole} \quad \text{(since } 40 \times 943 = 37720\text{)} \]
- Calculate the remainder:
\[ 37758 - 37720 = 38 \]
So,
\[ V = 943 \frac{38}{40} \]
Now, simplify \( \frac{38}{40} \):
\[ \frac{38 \div 2}{40 \div 2} = \frac{19}{20} \]
Thus, the volume in mixed number form is:
\[ 943 \frac{19}{20} , m^3 \]
So, the final answer is:
\[ \boxed{943 \frac{19}{20}} , m^3 \]