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

Did you know that the formulas for the volume of a cylinder and a sphere have some interesting relationships? For a cylinder, the volume \( V \) is calculated using the formula \( V = \pi r^2 h \), while for a sphere, it’s \( V = \frac{4}{3} \pi r^3 \). If the cylinder has a volume of \( 50 \mathrm{ft}^{3} \) and shares the same radius as the sphere, you can easily find the radius by rearranging the cylinder’s volume formula and plugging in the height! Now, if you ever find yourself grappling with understanding the difference in volume between shapes with the same dimensions, a nifty tip is to memorize the ratios! The volume of the sphere is \( \frac{2}{3} \) of the cylinder when they have the same height and radius. This means that if the cylinder is \( 50 \mathrm{ft}^3 \), the sphere will have a volume of about \( 33.33 \mathrm{ft}^3 \). Just think of it as the sphere being a little shy about filling the space compared to the cylinder!