Use the relationship among the formulas for the volumes of cones, cylinders, and spheres to solve the following problem. The volume of a cylinder is 36 cm3 . What is the volume of a sphere if its radius is the same as the cylinder’s and the height of the cylinder is equal to the sphere’s diameter?(1 point)

Question

54 cm3 54 cm cubed 24 cm3 24 cm cubed 18 cm3 18 cm cubed 12 cm3

Answer 1

Let's start by defining the given information:

1. The volume of the cylinder \( V_c = 36 , \text{cm}^3 \).
2. The height of the cylinder \( h \) is equal to the diameter of the sphere. The radius of the sphere (and cylinder) is \( r \).
3. The diameter of the sphere is \( d = 2r \).

The volume \( V_c \) of the cylinder is given by the formula: \[ V_c = \pi r^2 h \] Since the height \( h \) of the cylinder is equal to the diameter of the sphere: \[ h = 2r \] Substituting \( h \) in the cylinder volume formula: \[ V_c = \pi r^2 (2r) = 2\pi r^3 \] Now we know that \( V_c = 36 \): \[ 2\pi r^3 = 36 \] Dividing both sides by \( 2\pi \): \[ r^3 = \frac{36}{2\pi} = \frac{18}{\pi} \]

Next, we need to find the volume \( V_s \) of the sphere, which is given by the formula: \[ V_s = \frac{4}{3} \pi r^3 \] We can substitute \( r^3 \) from the previous equation: \[ V_s = \frac{4}{3} \pi \left(\frac{18}{\pi}\right) \] Simplifying this: \[ V_s = \frac{4}{3} \cdot 18 = \frac{72}{3} = 24 , \text{cm}^3 \]

Thus, the volume of the sphere is: \[ \boxed{24 , \text{cm}^3} \]