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 volume of 18 3/4 ft, a width of 3 ft, and a height of 1 1/4 ft, find its length
(b) The other crate has a base area of 7 1/2 ft, and a height of 2ft. Find its volume
(a) one crate has a volume of 18 3/4 ft, a width of 3 ft, and a height of 1 1/4 ft, find its length
(b) The other crate has a base area of 7 1/2 ft, and a height of 2ft. Find its volume
Answers
GPT-4o mini
To solve the problems, we'll use the formulas for volume and area of rectangular prisms.
### (a) Finding the length of the first crate
We know the volume \( V \) of a rectangular prism is given by:
\[
V = \text{length} \times \text{width} \times \text{height}
\]
For the first crate:
- Volume \( V = 18 \frac{3}{4} \, \text{ft}^3 = 18.75 \, \text{ft}^3 \) (convert the mixed number to an improper fraction or decimal)
- Width \( w = 3 \, \text{ft} \)
- Height \( h = 1 \frac{1}{4} \, \text{ft} = 1.25 \, \text{ft} \) (convert the mixed number to an improper fraction or decimal)
We can rearrange the volume formula to find the length \( l \):
\[
l = \frac{V}{w \times h}
\]
Substituting in the known values:
\[
l = \frac{18.75}{3 \times 1.25}
\]
Calculate the denominator:
\[
3 \times 1.25 = 3.75
\]
Now calculate the length:
\[
l = \frac{18.75}{3.75} = 5
\]
Thus, the length of the first crate is \( 5 \, \text{ft} \).
### (b) Finding the volume of the second crate
The volume \( V \) of a rectangular prism can also be calculated from the base area \( A \) and height \( h \):
\[
V = A \times h
\]
For the second crate:
- Base area \( A = 7 \frac{1}{2} \, \text{ft}^2 = 7.5 \, \text{ft}^2 \) (convert the mixed number to an improper fraction or decimal)
- Height \( h = 2 \, \text{ft} \)
Substituting in the known values:
\[
V = 7.5 \times 2
\]
Calculating the volume:
\[
V = 15 \, \text{ft}^3
\]
Thus, the volume of the second crate is \( 15 \, \text{ft}^3 \).
### Summary
- (a) The length of the first crate is \( 5 \, \text{ft} \).
- (b) The volume of the second crate is \( 15 \, \text{ft}^3 \).
### (a) Finding the length of the first crate
We know the volume \( V \) of a rectangular prism is given by:
\[
V = \text{length} \times \text{width} \times \text{height}
\]
For the first crate:
- Volume \( V = 18 \frac{3}{4} \, \text{ft}^3 = 18.75 \, \text{ft}^3 \) (convert the mixed number to an improper fraction or decimal)
- Width \( w = 3 \, \text{ft} \)
- Height \( h = 1 \frac{1}{4} \, \text{ft} = 1.25 \, \text{ft} \) (convert the mixed number to an improper fraction or decimal)
We can rearrange the volume formula to find the length \( l \):
\[
l = \frac{V}{w \times h}
\]
Substituting in the known values:
\[
l = \frac{18.75}{3 \times 1.25}
\]
Calculate the denominator:
\[
3 \times 1.25 = 3.75
\]
Now calculate the length:
\[
l = \frac{18.75}{3.75} = 5
\]
Thus, the length of the first crate is \( 5 \, \text{ft} \).
### (b) Finding the volume of the second crate
The volume \( V \) of a rectangular prism can also be calculated from the base area \( A \) and height \( h \):
\[
V = A \times h
\]
For the second crate:
- Base area \( A = 7 \frac{1}{2} \, \text{ft}^2 = 7.5 \, \text{ft}^2 \) (convert the mixed number to an improper fraction or decimal)
- Height \( h = 2 \, \text{ft} \)
Substituting in the known values:
\[
V = 7.5 \times 2
\]
Calculating the volume:
\[
V = 15 \, \text{ft}^3
\]
Thus, the volume of the second crate is \( 15 \, \text{ft}^3 \).
### Summary
- (a) The length of the first crate is \( 5 \, \text{ft} \).
- (b) The volume of the second crate is \( 15 \, \text{ft}^3 \).