Question
A cylinder shaped dispenser holds 5,652 cubic centimeters of liquid soap and is now full. The radius of the dispenser is 7.5 centimeters.
What is the difference between the height of the soap in the full dispenser and the height when 4,239 cubic centimeters of soap remains in the dispenser?
Use 3.14 for pi.
Enter your answer in the box.
What is the difference between the height of the soap in the full dispenser and the height when 4,239 cubic centimeters of soap remains in the dispenser?
Use 3.14 for pi.
Enter your answer in the box.
Answers
GPT-4o mini
To find the difference in height of the soap in the full dispenser and the height when 4,239 cubic centimeters of soap remains, we will follow these steps:
1. **Calculate the height of the full dispenser.**
The formula for the volume \( V \) of a cylinder is given by:
\[
V = \pi r^2 h
\]
where \( r \) is the radius and \( h \) is the height. First, we set the volume of the full dispenser to \( 5,652 \) cubic centimeters:
\[
5,652 = 3.14 \times (7.5)^2 \times h
\]
Calculating \( (7.5)^2 \):
\[
(7.5)^2 = 56.25
\]
Substituting this back into the volume formula:
\[
5,652 = 3.14 \times 56.25 \times h
\]
Now multiplying \( 3.14 \) and \( 56.25 \):
\[
3.14 \times 56.25 = 176.625
\]
So the equation becomes:
\[
5,652 = 176.625 \times h
\]
Solving for \( h \):
\[
h = \frac{5,652}{176.625} \approx 32
\]
2. **Calculate the height when there is 4,239 cubic centimeters of soap remaining.**
Using the same volume formula, set \( V = 4,239 \):
\[
4,239 = 3.14 \times 56.25 \times h
\]
This gives us:
\[
4,239 = 176.625 \times h
\]
Solving for \( h \):
\[
h = \frac{4,239}{176.625} \approx 24
\]
3. **Calculate the difference in height.**
The height difference between the full dispenser and the dispenser with 4,239 cubic centimeters of soap is:
\[
\text{Difference} = 32 - 24 = 8
\]
Thus, the difference between the height of the soap in the full dispenser and the height when 4,239 cubic centimeters of soap remains in the dispenser is **8 centimeters**.
1. **Calculate the height of the full dispenser.**
The formula for the volume \( V \) of a cylinder is given by:
\[
V = \pi r^2 h
\]
where \( r \) is the radius and \( h \) is the height. First, we set the volume of the full dispenser to \( 5,652 \) cubic centimeters:
\[
5,652 = 3.14 \times (7.5)^2 \times h
\]
Calculating \( (7.5)^2 \):
\[
(7.5)^2 = 56.25
\]
Substituting this back into the volume formula:
\[
5,652 = 3.14 \times 56.25 \times h
\]
Now multiplying \( 3.14 \) and \( 56.25 \):
\[
3.14 \times 56.25 = 176.625
\]
So the equation becomes:
\[
5,652 = 176.625 \times h
\]
Solving for \( h \):
\[
h = \frac{5,652}{176.625} \approx 32
\]
2. **Calculate the height when there is 4,239 cubic centimeters of soap remaining.**
Using the same volume formula, set \( V = 4,239 \):
\[
4,239 = 3.14 \times 56.25 \times h
\]
This gives us:
\[
4,239 = 176.625 \times h
\]
Solving for \( h \):
\[
h = \frac{4,239}{176.625} \approx 24
\]
3. **Calculate the difference in height.**
The height difference between the full dispenser and the dispenser with 4,239 cubic centimeters of soap is:
\[
\text{Difference} = 32 - 24 = 8
\]
Thus, the difference between the height of the soap in the full dispenser and the height when 4,239 cubic centimeters of soap remains in the dispenser is **8 centimeters**.