The volume of a sphere is 36 m^3. What is the volume of a cylinder if it's radius is the same as the sphere's and it's height is equal to the sphere's diameter? Round to the nearest whole number

Answer 1

To find the volume of the cylinder, we first need to understand the relationship between the volume of the sphere and its dimensions.

1. Volume of the Sphere: The formula for the volume \( V \) of a sphere is given by:

   \[ V = \frac{4}{3} \pi r^3 \]

   Given that the volume of the sphere is \( 36 , \text{m}^3 \), we can set up the equation:

   \[ \frac{4}{3} \pi r^3 = 36 \]

   To solve for the radius \( r \), we rearrange the equation:

   \[ r^3 = \frac{36 \times 3}{4 \pi} \]

   Simplifying this:

   \[ r^3 = \frac{108}{4 \pi} = \frac{27}{\pi} \]

   Taking the cube root:

   \[ r = \left( \frac{27}{\pi} \right)^{\frac{1}{3}} \]

2. Finding the Height of the Cylinder: The height \( h \) of the cylinder is equal to the diameter of the sphere. The diameter \( d \) is given by:

   \[ d = 2r \]

   Thus, the height of the cylinder is:

   \[ h = 2r = 2 \left( \frac{27}{\pi} \right)^{\frac{1}{3}} \]

3. Volume of the Cylinder: The formula for the volume \( V_c \) of a cylinder is:

   \[ V_c = \pi r^2 h \]

   Substituting \( h = 2r \):

   \[ V_c = \pi r^2 (2r) = 2 \pi r^3 \]

   We already have \( r^3 \):

   \[ V_c = 2 \pi \left( \frac{27}{\pi} \right) = 2 \times 27 = 54 , \text{m}^3 \]

Thus, the volume of the cylinder is:

\[ \boxed{54 , \text{m}^3} \]