To find the volume of a rectangular prism, we use the formula:
\[ \text{Volume} = \text{length} \times \text{width} \times \text{height} \]
First, we convert the mixed numbers to improper fractions:
-
Length: \(3 \frac{2}{3}\)
- Convert: \(3 \times 3 + 2 = 9 + 2 = 11\)
- Therefore, \(3 \frac{2}{3} = \frac{11}{3}\)
-
Height: \(4 \frac{1}{3}\)
- Convert: \(4 \times 3 + 1 = 12 + 1 = 13\)
- Therefore, \(4 \frac{1}{3} = \frac{13}{3}\)
-
Width: \(2 \frac{1}{4}\)
- Convert: \(2 \times 4 + 1 = 8 + 1 = 9\)
- Therefore, \(2 \frac{1}{4} = \frac{9}{4}\)
Now, we can calculate the volume:
\[ \text{Volume} = \frac{11}{3} \times \frac{13}{3} \times \frac{9}{4} \]
First, multiply the numerators:
\[ 11 \times 13 \times 9 = 1287 \]
Next, multiply the denominators:
\[ 3 \times 3 \times 4 = 36 \]
Thus, we have:
\[ \text{Volume} = \frac{1287}{36} \]
Now, simplify \(\frac{1287}{36}\):
To simplify, we need to divide both the numerator and denominator by their greatest common divisor (gcd). The \(gcd(1287, 36)\) can be found as follows:
-
Prime factorization of \(36 = 2^2 \times 3^2\).
-
Check divisibility of \(1287\) by \(3\):
- Sum of the digits: \(1 + 2 + 8 + 7 = 18\) (which is divisible by \(3\)): \[ 1287 \div 3 = 429 \]
Check \(429\): \[ 4 + 2 + 9 = 15 \quad (divisible , by , 3) \Rightarrow 429 \div 3 = 143 \] \(143 = 11 \times 13\)
Thus, we find that \(1287 = 3^2 \times 11 \times 13\).
Now, simplification:
\[ \frac{1287}{36} = \frac{3^2 \times 11 \times 13}{2^2 \times 3^2} = \frac{11 \times 13}{2^2} = \frac{143}{4} \]
Finally, express \(\frac{143}{4}\) as a mixed number:
\[ 143 \div 4 = 35 \quad \text{with a remainder of } 3 \]
So, the mixed number is:
\[ 35 \frac{3}{4} \]
Thus, the volume of the rectangular prism is:
\[ \boxed{\frac{143}{4}} \quad \text{or } 35 \frac{3}{4} \text{ ft}^3 \]