Question
A sphere and a cylinder have the same radius and height. The volume of the cylinder is 18 centimeters cubed.
A sphere with height h and radius r. A cylinder with height h and radius r.
What is the volume of the sphere?
12 centimeters cubed
24 centimeters cubed
36 centimeters cubed
54 centimeters cubed
A sphere with height h and radius r. A cylinder with height h and radius r.
What is the volume of the sphere?
12 centimeters cubed
24 centimeters cubed
36 centimeters cubed
54 centimeters cubed
Answers
GPT-4o mini
To find the volume of the sphere, we first need to establish the relationship between the volume of the cylinder and the sphere.
The volume of a cylinder is given by the formula:
\[
V_{cylinder} = \pi r^2 h
\]
According to the problem, the volume of the cylinder is 18 cm³:
\[
\pi r^2 h = 18
\]
The volume of a sphere is given by the formula:
\[
V_{sphere} = \frac{4}{3} \pi r^3
\]
Since the height and radius of both the sphere and the cylinder are the same, we can use the height of the cylinder to find the radius in terms of the volume of the cylinder.
From the cylinder's volume equation, we have:
\[
h = \frac{18}{\pi r^2}
\]
Since the height of the sphere is equal to the height of the cylinder, we have \( h = 2r \) for the sphere (the height of a sphere can be considered double its radius for this case).
Substituting \( h \) into our equation:
\[
2r = \frac{18}{\pi r^2}
\]
Now, we will solve for the radius \( r \). Multiplying both sides by \( \pi r^2 \) gives:
\[
2\pi r^3 = 18
\]
Now dividing both sides by 2:
\[
\pi r^3 = 9
\]
Now, we can solve for \( r^3 \):
\[
r^3 = \frac{9}{\pi}
\]
Next, we will substitute this value of \( r \) back into the volume formula for the sphere to find its volume:
\[
V_{sphere} = \frac{4}{3} \pi r^3
\]
Substituting \( r^3 \) with \( \frac{9}{\pi} \):
\[
V_{sphere} = \frac{4}{3} \pi \left(\frac{9}{\pi}\right)
\]
Simplifying this:
\[
V_{sphere} = \frac{4}{3} \cdot 9
\]
\[
V_{sphere} = \frac{36}{3} = 12 \, \text{cm}^3
\]
Thus, the volume of the sphere is:
\[
\boxed{12 \, \text{cm}^3}
\]
The volume of a cylinder is given by the formula:
\[
V_{cylinder} = \pi r^2 h
\]
According to the problem, the volume of the cylinder is 18 cm³:
\[
\pi r^2 h = 18
\]
The volume of a sphere is given by the formula:
\[
V_{sphere} = \frac{4}{3} \pi r^3
\]
Since the height and radius of both the sphere and the cylinder are the same, we can use the height of the cylinder to find the radius in terms of the volume of the cylinder.
From the cylinder's volume equation, we have:
\[
h = \frac{18}{\pi r^2}
\]
Since the height of the sphere is equal to the height of the cylinder, we have \( h = 2r \) for the sphere (the height of a sphere can be considered double its radius for this case).
Substituting \( h \) into our equation:
\[
2r = \frac{18}{\pi r^2}
\]
Now, we will solve for the radius \( r \). Multiplying both sides by \( \pi r^2 \) gives:
\[
2\pi r^3 = 18
\]
Now dividing both sides by 2:
\[
\pi r^3 = 9
\]
Now, we can solve for \( r^3 \):
\[
r^3 = \frac{9}{\pi}
\]
Next, we will substitute this value of \( r \) back into the volume formula for the sphere to find its volume:
\[
V_{sphere} = \frac{4}{3} \pi r^3
\]
Substituting \( r^3 \) with \( \frac{9}{\pi} \):
\[
V_{sphere} = \frac{4}{3} \pi \left(\frac{9}{\pi}\right)
\]
Simplifying this:
\[
V_{sphere} = \frac{4}{3} \cdot 9
\]
\[
V_{sphere} = \frac{36}{3} = 12 \, \text{cm}^3
\]
Thus, the volume of the sphere is:
\[
\boxed{12 \, \text{cm}^3}
\]