Question

What is the volume, in cubic inches, of a rectangular prism with a height of 3 2/3 in., a length of 2 1/6 in., and a width of 1 1/2 in.? Your answer should be a mixed number in simplest form. (1 point) (3 2/3 is a fraction, along with 2 1/6 and 1 1/2)

Answer 1

To find the volume of a rectangular prism, we use the formula:

\[
\text{Volume} = \text{length} \times \text{width} \times \text{height}
\]

First, let's convert the mixed numbers into improper fractions.

1. For the height \(3 \frac{2}{3}\):
\[
3 \frac{2}{3} = \frac{3 \times 3 + 2}{3} = \frac{9 + 2}{3} = \frac{11}{3}
\]

2. For the length \(2 \frac{1}{6}\):
\[
2 \frac{1}{6} = \frac{2 \times 6 + 1}{6} = \frac{12 + 1}{6} = \frac{13}{6}
\]

3. For the width \(1 \frac{1}{2}\):
\[
1 \frac{1}{2} = \frac{1 \times 2 + 1}{2} = \frac{2 + 1}{2} = \frac{3}{2}
\]

Now we can substitute the improper fractions into the volume formula:

\[
\text{Volume} = \left(\frac{11}{3}\right) \times \left(\frac{13}{6}\right) \times \left(\frac{3}{2}\right)
\]

Next, we calculate the product step by step:

1. Multiply \(\frac{11}{3}\) and \(\frac{13}{6}\):
\[
\frac{11 \times 13}{3 \times 6} = \frac{143}{18}
\]

2. Then multiply by \(\frac{3}{2}\):
\[
\frac{143}{18} \times \frac{3}{2} = \frac{143 \times 3}{18 \times 2} = \frac{429}{36}
\]

Next, we simplify \(\frac{429}{36}\). We need to convert this improper fraction into a mixed number.

1. Divide \(429\) by \(36\):
\[
429 \div 36 = 11 \quad \text{(because } 36 \times 11 = 396\text{)}
\]
The remainder is:
\[
429 - 396 = 33
\]
So, we can express \(\frac{429}{36}\) as \(11\) and a remainder of \(33\) over \(36\).

This gives us:
\[
\frac{429}{36} = 11 \frac{33}{36}
\]

Next, we simplify \(\frac{33}{36}\) by finding the GCD, which is \(3\):
\[
\frac{33 \div 3}{36 \div 3} = \frac{11}{12}
\]

Thus, the volume of the rectangular prism in simplest mixed number form is:
\[
\boxed{11 \frac{11}{12}}
\]