A sphere and a cylinder have the same radius and height. The volume of the cylinder is $$ 50 \mathrm{ft}^{2} $$.

Question

UpStudy AI · Accepted Answer

Let the common radius be $$ r $$ and note that for a sphere the height is its diameter. Therefore, the cylinder’s height is  
$$
h = 2r.
$$

The volume of the cylinder is given by  
$$
V_{\text{cyl}} = \pi r^2 h.
$$
Substituting $$ h = 2r $$ gives  
$$
V_{\text{cyl}} = \pi r^2 (2r) = 2\pi r^3.
$$
We are told that  
$$
2\pi r^3 = 50.
$$
Solving for $$ r^3 $$ yields  
$$
r^3 = \frac{50}{2\pi} = \frac{25}{\pi}.
$$

The volume of a sphere is  
$$
V_{\text{sphere}} = \frac{4}{3}\pi r^3.
$$
Substitute $$ r^3 = \frac{25}{\pi} $$ into the sphere’s volume formula:  
$$
V_{\text{sphere}} = \frac{4}{3}\pi \left(\frac{25}{\pi}\right) = \frac{4}{3} \times 25 = \frac{100}{3}.
$$

Thus, the volume of the sphere is  
$$
\frac{100}{3} \, \text{ft}^3.
$$
The volume of the sphere is $$ \frac{100}{3} $$ cubic feet.

A sphere and a cylinder have the same radius and height. The volume of the cylinder is \( 50 \mathrm{ft}^{2} \).

Upstudy AI Solution

Answer

Solution

Mind Expander

Related Questions

Latest Geometry Questions