Question

A pet shop sells two crates with different dimensions. Each crate is shaped like a rectangular prism. Use the given information to answer each part below.
(a)One crate has a length of 5 3/4, a width of 3, and a height of 4. Find its volume.

(b)The other crate has a volume of 22 1/2 cubed and a base area of 15 squared. Find its height.

Answer 1

To solve these problems, we need to use the formula for the volume of a rectangular prism, which is given by:

\[
\text{Volume} = \text{Length} \times \text{Width} \times \text{Height}
\]

### (a) Finding the Volume of the First Crate

1. **Convert Mixed Numbers to Improper Fractions:**
- Length: \(5 \frac{3}{4} = \frac{5 \times 4 + 3}{4} = \frac{20 + 3}{4} = \frac{23}{4}\)
- Width: \(3\) is already a whole number.
- Height: \(4\) is also a whole number.

2. **Calculate the Volume:**
\[
\text{Volume} = \text{Length} \times \text{Width} \times \text{Height} = \frac{23}{4} \times 3 \times 4
\]
Here, we can simplify before multiplying:
\[
\frac{23}{4} \cdot 4 = 23
\]
Thus:
\[
\text{Volume} = 23 \times 3 = 69
\]
Therefore, the volume of the first crate is \(69\) cubic units.

### (b) Finding the Height of the Second Crate

1. **Use the Volume Formula:**
Given:
- Volume = \(22 \frac{1}{2} = \frac{45}{2}\) (converting to an improper fraction)
- Base Area = \(15\)

2. **Volume Formula Rearrangement:**
\[
\text{Volume} = \text{Base Area} \times \text{Height}
\]
Rearranging for height:
\[
\text{Height} = \frac{\text{Volume}}{\text{Base Area}} = \frac{\frac{45}{2}}{15}
\]

3. **Simplifying:**
\[
\text{Height} = \frac{45}{2} \div 15 = \frac{45}{2} \times \frac{1}{15} = \frac{45}{30} = \frac{3}{2} = 1 \frac{1}{2}
\]

Thus, the height of the second crate is \(1 \frac{1}{2}\) units.

### Summary:
- (a) The volume of the first crate is \(69\) cubic units.
- (b) The height of the second crate is \(1 \frac{1}{2}\) units.