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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...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
30 inches
30 inches
15 inches
15 inches
5 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
30 inches
30 inches
15 inches
15 inches
5 inches
Answers
GPT-4o mini
To find the diameter of the cylinder, 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, and \( h \) is the height.
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:
\[
h = 42 \text{ cm} \times \frac{1 \text{ in}}{2.54 \text{ cm}} \approx 16.54 \text{ in}
\]
Now, we can rearrange the volume formula to solve for the radius \( r \):
\[
r^2 = \frac{V}{\pi h}
\]
Substituting the volume and height into the equation:
\[
r^2 = \frac{29673}{3.14 \times 16.54}
\]
Calculating the denominator:
\[
3.14 \times 16.54 \approx 51.9276
\]
Now calculating \( r^2 \):
\[
r^2 = \frac{29673}{51.9276} \approx 571.64
\]
Taking the square root to find \( r \):
\[
r \approx \sqrt{571.64} \approx 23.9 \text{ inches}
\]
Now, to find the diameter \( d \):
\[
d = 2r \approx 2 \times 23.9 \approx 47.8 \text{ inches}
\]
Since none of the multiple-choice options (10 inches, 30 inches, 15 inches, 5 inches) matches the calculated diameter, it seems there might be a misinterpretation of the problem or wrong data provided. Please verify the height or volume details, or if calculations yield an unexpected result.
\[
V = \pi r^2 h
\]
where \( V \) is the volume, \( r \) is the radius, and \( h \) is the height.
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:
\[
h = 42 \text{ cm} \times \frac{1 \text{ in}}{2.54 \text{ cm}} \approx 16.54 \text{ in}
\]
Now, we can rearrange the volume formula to solve for the radius \( r \):
\[
r^2 = \frac{V}{\pi h}
\]
Substituting the volume and height into the equation:
\[
r^2 = \frac{29673}{3.14 \times 16.54}
\]
Calculating the denominator:
\[
3.14 \times 16.54 \approx 51.9276
\]
Now calculating \( r^2 \):
\[
r^2 = \frac{29673}{51.9276} \approx 571.64
\]
Taking the square root to find \( r \):
\[
r \approx \sqrt{571.64} \approx 23.9 \text{ inches}
\]
Now, to find the diameter \( d \):
\[
d = 2r \approx 2 \times 23.9 \approx 47.8 \text{ inches}
\]
Since none of the multiple-choice options (10 inches, 30 inches, 15 inches, 5 inches) matches the calculated diameter, it seems there might be a misinterpretation of the problem or wrong data provided. Please verify the height or volume details, or if calculations yield an unexpected result.