Question
The volume of a right circular cylinder is 29,673 inches3
. Find the diameter of the cylinder if the height is 42 cm. Use 3.14 for π
.(1 point)
Responses
10 inches
10 inches
5 inches
5 inches
30 inches
30 inches
15 inches
. Find the diameter of the cylinder if the height is 42 cm. Use 3.14 for π
.(1 point)
Responses
10 inches
10 inches
5 inches
5 inches
30 inches
30 inches
15 inches
Answers
GPT-4o mini
To find the diameter of a right circular cylinder given its 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,
- \( \pi \) is approximately 3.14.
We know the volume \( V = 29,673 \) cubic inches and the height \( h = 42 \) cm. However, we need to convert the height from centimeters to inches since the volume is in cubic inches.
To convert centimeters to inches, we use the conversion factor:
\[
1 \text{ inch} = 2.54 \text{ cm}
\]
Thus,
\[
h = 42 \text{ cm} \times \frac{1 \text{ inch}}{2.54 \text{ cm}} \approx 16.54 \text{ inches}
\]
Now we can substitute the values into the volume formula:
\[
29,673 = 3.14 r^2 (16.54)
\]
Next, we can isolate \( r^2 \):
\[
r^2 = \frac{29,673}{3.14 \times 16.54}
\]
Calculating the denominator:
\[
3.14 \times 16.54 \approx 51.9006
\]
Now, substituting this back into the equation gives:
\[
r^2 = \frac{29,673}{51.9006} \approx 571.52
\]
Next, taking the square root to find \( r \):
\[
r \approx \sqrt{571.52} \approx 23.93 \text{ inches}
\]
Finally, the diameter \( d \) is twice the radius:
\[
d = 2r \approx 2 \times 23.93 \approx 47.86 \text{ inches}
\]
None of the provided responses match this calculation, but we can check:
- 10 inches
- 5 inches
- 30 inches
- 15 inches
The calculated diameter does not align with any of the options provided. Please recheck the values for upward verification or the overall context of the question. However, based on the volume and height given, the diameter based on the mathematical calculations does not yield an appropriate response in the options available.
\[
V = \pi r^2 h
\]
Where:
- \( V \) is the volume,
- \( r \) is the radius,
- \( h \) is the height,
- \( \pi \) is approximately 3.14.
We know the volume \( V = 29,673 \) cubic inches and the height \( h = 42 \) cm. However, we need to convert the height from centimeters to inches since the volume is in cubic inches.
To convert centimeters to inches, we use the conversion factor:
\[
1 \text{ inch} = 2.54 \text{ cm}
\]
Thus,
\[
h = 42 \text{ cm} \times \frac{1 \text{ inch}}{2.54 \text{ cm}} \approx 16.54 \text{ inches}
\]
Now we can substitute the values into the volume formula:
\[
29,673 = 3.14 r^2 (16.54)
\]
Next, we can isolate \( r^2 \):
\[
r^2 = \frac{29,673}{3.14 \times 16.54}
\]
Calculating the denominator:
\[
3.14 \times 16.54 \approx 51.9006
\]
Now, substituting this back into the equation gives:
\[
r^2 = \frac{29,673}{51.9006} \approx 571.52
\]
Next, taking the square root to find \( r \):
\[
r \approx \sqrt{571.52} \approx 23.93 \text{ inches}
\]
Finally, the diameter \( d \) is twice the radius:
\[
d = 2r \approx 2 \times 23.93 \approx 47.86 \text{ inches}
\]
None of the provided responses match this calculation, but we can check:
- 10 inches
- 5 inches
- 30 inches
- 15 inches
The calculated diameter does not align with any of the options provided. Please recheck the values for upward verification or the overall context of the question. However, based on the volume and height given, the diameter based on the mathematical calculations does not yield an appropriate response in the options available.