Question
The volume of a right circular cylinder is 29,673 inches3
29
,
673
inches
3
. Find the diameter of the cylinder if the height is 42 cm. Use 3.14 for π
𝜋
.(1 point)
Responses
10 inches
10 inches
5 inches
5 inches
15 inches
15 inches
30 inches
29
,
673
inches
3
. Find the diameter of the cylinder if the height is 42 cm. Use 3.14 for π
𝜋
.(1 point)
Responses
10 inches
10 inches
5 inches
5 inches
15 inches
15 inches
30 inches
Answers
GPT-4o mini
To find the diameter of the right circular cylinder given the volume and height, we can use the formula for the volume of a cylinder:
\[
V = \pi r^2 h
\]
Where:
- \( V \) is the volume,
- \( r \) is the radius,
- \( h \) is the height.
Given:
- \( V = 29,673 \, \text{inches}^3 \)
- \( h = 42 \, \text{cm} \)
First, we need to convert the height from centimeters to inches since the volume is given in cubic inches. There are 2.54 centimeters in an inch, so:
\[
h = 42 \, \text{cm} \times \frac{1 \, \text{inch}}{2.54 \, \text{cm}} \approx 16.54 \, \text{inches}
\]
Now we can substitute the volume and height into the volume formula:
\[
29,673 = 3.14 r^2 (16.54)
\]
Let's solve for \( r^2 \):
\[
29,673 = 51.8756 r^2
\]
Now, divide both sides by \( 51.8756 \):
\[
r^2 = \frac{29,673}{51.8756} \approx 571.57
\]
Now, take the square root of both sides to find \( r \):
\[
r \approx \sqrt{571.57} \approx 23.92 \, \text{inches}
\]
Finally, to find the diameter \( d \), we use:
\[
d = 2r \approx 2 \times 23.92 \approx 47.84 \, \text{inches}
\]
It seems there was a mistake or misunderstanding in the provided options as none of them match. Therefore, if you're looking for the closest response you can consider the possible physical context but based on calculations, the diameter is approximately \( 47.84 \, \text{inches} \). If we only choose from the options given, they may not represent correct calculations based on the provided volume and height.
\[
V = \pi r^2 h
\]
Where:
- \( V \) is the volume,
- \( r \) is the radius,
- \( h \) is the height.
Given:
- \( V = 29,673 \, \text{inches}^3 \)
- \( h = 42 \, \text{cm} \)
First, we need to convert the height from centimeters to inches since the volume is given in cubic inches. There are 2.54 centimeters in an inch, so:
\[
h = 42 \, \text{cm} \times \frac{1 \, \text{inch}}{2.54 \, \text{cm}} \approx 16.54 \, \text{inches}
\]
Now we can substitute the volume and height into the volume formula:
\[
29,673 = 3.14 r^2 (16.54)
\]
Let's solve for \( r^2 \):
\[
29,673 = 51.8756 r^2
\]
Now, divide both sides by \( 51.8756 \):
\[
r^2 = \frac{29,673}{51.8756} \approx 571.57
\]
Now, take the square root of both sides to find \( r \):
\[
r \approx \sqrt{571.57} \approx 23.92 \, \text{inches}
\]
Finally, to find the diameter \( d \), we use:
\[
d = 2r \approx 2 \times 23.92 \approx 47.84 \, \text{inches}
\]
It seems there was a mistake or misunderstanding in the provided options as none of them match. Therefore, if you're looking for the closest response you can consider the possible physical context but based on calculations, the diameter is approximately \( 47.84 \, \text{inches} \). If we only choose from the options given, they may not represent correct calculations based on the provided volume and height.