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)
Responses
12 cm3
12 cm cubed
54 cm3
54 cm cubed
24 cm3
24 cm cubed
18 cm3
Given:
- The volume of the cylinder is \( V_{cylinder} = 36 \text{ cm}^3 \).
- The height of the cylinder is equal to the sphere's diameter.
- The radius of the cylinder is the same as the radius of the sphere.
First, let's express the volume of the cylinder with the formula:
\[ V_{cylinder} = \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height.
The volume of the cylinder is already given as \( 36 \text{ cm}^3 \). The height of the cylinder is twice the radius of the cylinder (since the height is the sphere's diameter, which is two times the radius of a sphere with the same radius as the cylinder).
So we can deduce that:
\[ h = 2r \]
Given the known volume, we can solve for the radius:
\[ 36 = \pi r^2 (2r) \]
\[ 36 = 2\pi r^3 \]
\[ r^3 = \frac{36}{2\pi} \]
\[ r^3 = \frac{18}{\pi} \]
Now, let's find the volume of a sphere using the formula:
\[ V_{sphere} = \frac{4}{3} \pi r^3 \]
Substitute the value of \( r^3 \) we've found into the volume formula for the sphere:
\[ V_{sphere} = \frac{4}{3} \pi \left( \frac{18}{\pi} \right) \]
Simplify:
\[ V_{sphere} = \frac{4}{3} \times 18 \]
\[ V_{sphere} = 4 \times 6 \]
\[ V_{sphere} = 24 \text{ cm}^3 \]
The volume of the sphere is therefore \( 24 \text{ cm}^3 \).
The correct answer is 24 cm³ (or 24 cubic centimeters).