Volume Of Right Rectangular Prisms, Free PDF Download Learn, 55% OFF - Free Printable

To solve the problem of finding the volumes of the given shapes, we will use the formulas for the volume of a rectangular prism and a cube. Let's go through each part step by step.

Part 1: Rectangular Prisms

#### 1. First Rectangular Prism
- Dimensions: \(2 \frac{2}{3} \text{ in.}\), \(3 \frac{1}{2} \text{ in.}\), \(6 \text{ in.}\)

First, convert the mixed numbers to improper fractions:
- \(2 \frac{2}{3} = \frac{8}{3}\)
- \(3 \frac{1}{2} = \frac{7}{2}\)

The volume \(V\) of a rectangular prism is given by:
\[ V = \text{length} \times \text{width} \times \text{height} \]

Substitute the values:
\[ V = \frac{8}{3} \times \frac{7}{2} \times 6 \]

First, simplify the multiplication:
\[ V = \frac{8 \times 7 \times 6}{3 \times 2} = \frac{336}{6} = 56 \]

So, the volume is:
\[ \boxed{56} \text{ cubic inches} \]

#### 2. Second Rectangular Prism
- Dimensions: \(\frac{3}{4} \text{ in.}\), \(\frac{1}{3} \text{ in.}\), \(1 \frac{1}{5} \text{ in.}\)

Convert the mixed number to an improper fraction:
- \(1 \frac{1}{5} = \frac{6}{5}\)

The volume \(V\) is:
\[ V = \frac{3}{4} \times \frac{1}{3} \times \frac{6}{5} \]

Simplify the multiplication:
\[ V = \frac{3 \times 1 \times 6}{4 \times 3 \times 5} = \frac{18}{60} = \frac{3}{10} \]

So, the volume is:
\[ \boxed{\frac{3}{10}} \text{ cubic inches} \]

#### 3. Third Rectangular Prism
- Dimensions: \(\frac{1}{2} \text{ in.}\), \(\frac{1}{2} \text{ in.}\), \(2 \frac{3}{4} \text{ in.}\)

Convert the mixed number to an improper fraction:
- \(2 \frac{3}{4} = \frac{11}{4}\)

The volume \(V\) is:
\[ V = \frac{1}{2} \times \frac{1}{2} \times \frac{11}{4} \]

Simplify the multiplication:
\[ V = \frac{1 \times 1 \times 11}{2 \times 2 \times 4} = \frac{11}{16} \]

So, the volume is:
\[ \boxed{\frac{11}{16}} \text{ cubic inches} \]

Part 2: Cubes

#### 1. Cube with Edge Length \(\frac{1}{6}\)
The volume \(V\) of a cube is given by:
\[ V = \text{edge length}^3 \]

Substitute the edge length:
\[ V = \left(\frac{1}{6}\right)^3 = \frac{1}{6} \times \frac{1}{6} \times \frac{1}{6} = \frac{1}{216} \]

So, the volume is:
\[ \boxed{\frac{1}{216}} \text{ cubic units} \]

#### 2. Cube with Edge Length \(\frac{2}{3}\)
Substitute the edge length:
\[ V = \left(\frac{2}{3}\right)^3 = \frac{2}{3} \times \frac{2}{3} \times \frac{2}{3} = \frac{8}{27} \]

So, the volume is:
\[ \boxed{\frac{8}{27}} \text{ cubic units} \]

#### 3. Cube with Edge Length \(\frac{3}{4}\)
Substitute the edge length:
\[ V = \left(\frac{3}{4}\right)^3 = \frac{3}{4} \times \frac{3}{4} \times \frac{3}{4} = \frac{27}{64} \]

So, the volume is:
\[ \boxed{\frac{27}{64}} \text{ cubic units} \]

Final Answers

1. \( \boxed{56} \)
2. \( \boxed{\frac{3}{10}} \)
3. \( \boxed{\frac{11}{16}} \)
4. \( \boxed{\frac{1}{216}} \)
5. \( \boxed{\frac{8}{27}} \)
6. \( \boxed{\frac{27}{64}} \)