To find the volume of a sphere given that it has the same radius and height (which is equal to the diameter) as the cylinder, we can follow these steps:
Step 1: Understand the relationship between the cylinder and the sphere.
- The cylinder has a volume [tex]\( V_{\text{cylinder}} = 18 \, m^3 \)[/tex].
- The volume of a cylinder is given by the formula [tex]\( V = \pi r^2 h \)[/tex].
- The height of the cylinder is double the radius (since the height of the cylinder equals the diameter of the sphere), so we can write [tex]\( h = 2r \)[/tex].
Step 2: Express the volume of the cylinder in terms of the radius.
- Using the cylinder's volume formula and the relationship [tex]\( h = 2r \)[/tex], we get:
[tex]\[ V_{\text{cylinder}} = \pi r^2 \cdot 2r = 2\pi r^3 \][/tex]
- We know [tex]\( V_{\text{cylinder}} = 18 \, m^3 \)[/tex], so:
[tex]\[ 18 = 2\pi r^3 \][/tex]
Step 3: Solve for the radius [tex]\( r \)[/tex].
- Rearrange the equation to isolate [tex]\( r^3 \)[/tex]:
[tex]\[ r^3 = \frac{18}{2\pi} \][/tex]
[tex]\[ r^3 = \frac{9}{\pi} \][/tex]
Step 4: Calculate the radius cubed.
- The specific value is:
[tex]\[ r^3 \approx 2.8648 \][/tex]
Step 5: Find the volume of the sphere.
- The volume of a sphere is given by [tex]\( V = \frac{4}{3} \pi r^3 \)[/tex].
- Substitute [tex]\( r^3 \)[/tex]:
[tex]\[ V_{\text{sphere}} = \frac{4}{3} \pi \cdot \frac{9}{\pi} \][/tex]
- Simplify the expression:
[tex]\[ V_{\text{sphere}} = \frac{4}{3} \cdot 9 \][/tex]
[tex]\[ V_{\text{sphere}} = 12 \, m^3 \][/tex]
Therefore, the volume of the sphere is [tex]\( 12 \, m^3 \)[/tex].
