A cylinder and a sphere shown below have equal volumes what is the radius of the sphere

A cylinder and a sphere shown below have equal volumes what is the radius of the sphere

The Correct answer and Explanation is:

To solve for the radius of the sphere when the volumes of the cylinder and sphere are equal, we start by using the volume formulas for both shapes.

Step 1: Volume Formulas

• The volume of a cylinder VcylinderV_{\text{cylinder}}Vcylinder​ is given by:Vcylinder=πrcylinder2hV_{\text{cylinder}} = \pi r_{\text{cylinder}}^2 hVcylinder​=πrcylinder2​hwhere rcylinderr_{\text{cylinder}}rcylinder​ is the radius of the cylinder’s base and hhh is the height of the cylinder.
• The volume of a sphere VsphereV_{\text{sphere}}Vsphere​ is given by:Vsphere=43πrsphere3V_{\text{sphere}} = \frac{4}{3} \pi r_{\text{sphere}}^3Vsphere​=34​πrsphere3​where rspherer_{\text{sphere}}rsphere​ is the radius of the sphere.

Step 2: Equating the Volumes

Since the volumes are equal, we set the two formulas equal to each other:πrcylinder2h=43πrsphere3\pi r_{\text{cylinder}}^2 h = \frac{4}{3} \pi r_{\text{sphere}}^3πrcylinder2​h=34​πrsphere3​

Step 3: Simplify the Equation

Cancel out the common π\piπ term on both sides:rcylinder2h=43rsphere3r_{\text{cylinder}}^2 h = \frac{4}{3} r_{\text{sphere}}^3rcylinder2​h=34​rsphere3​

Next, solve for rsphere3r_{\text{sphere}}^3rsphere3​:rsphere3=3rcylinder2h4r_{\text{sphere}}^3 = \frac{3 r_{\text{cylinder}}^2 h}{4}rsphere3​=43rcylinder2​h​

Finally, take the cube root to solve for rspherer_{\text{sphere}}rsphere​:rsphere=(3rcylinder2h4)1/3r_{\text{sphere}} = \left( \frac{3 r_{\text{cylinder}}^2 h}{4} \right)^{1/3}rsphere​=(43rcylinder2​h​)1/3

Step 4: Interpretation

The radius of the sphere depends on the radius of the cylinder’s base rcylinderr_{\text{cylinder}}rcylinder​ and the height hhh of the cylinder. If you have specific values for rcylinderr_{\text{cylinder}}rcylinder​ and hhh, you can substitute them into the formula to find the exact radius of the sphere. The cube root calculation ensures the sphere has the same volume as the cylinder.

This result demonstrates how the shape of an object affects its volume, as a sphere requires a different radius than a cylinder to enclose the same amount of space.