cylinder is 36 cm^3. what is the volume of a sphere if its radius is the same as the cylinders and the height of the cylinder is equal to the spheres diameter?

Question

-12 cm^3
-54 cm^3
-24 cm^3
-18 cm^3

Answer 1

To find the volume of the sphere with the given conditions, let's first find the radius of the cylinder.

The volume \( V \) of a cylinder is given by the formula:

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

where \( r \) is the radius and \( h \) is the height of the cylinder. We know from the problem that the volume of the cylinder \( V = 36 , \text{cm}^3 \).

Next, let's denote the height of the cylinder as \( h \). According to the problem, the radius of the cylinder is equal to the radius of the sphere, and the height of the cylinder is equal to the diameter of the sphere, which is \( 2r \).

Thus, we can rewrite the equation for the volume of the cylinder using these relationships:

\[ 36 = \pi r^2 (2r) \]

This simplifies to:

\[ 36 = 2 \pi r^3 \]

Now, we can solve for \( r^3 \):

\[ r^3 = \frac{36}{2 \pi} = \frac{18}{\pi} \]

Next, we calculate the volume \( V_s \) of the sphere, which is given by the formula:

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

Substituting \( r^3 \) into the sphere’s volume formula:

\[ V_s = \frac{4}{3} \pi \left(\frac{18}{\pi}\right) \]

This simplifies to:

\[ V_s = \frac{4}{3} \cdot 18 = 24 , \text{cm}^3 \]

Thus, the volume of the sphere is \(\boxed{24}, \text{cm}^3\).

Answer 2

The volume of a cone is 25/3 pi cm^3. what is the volume of a sphere if its radius is the same as the cones and the height of the cone is equal to the spheres diameter
- 50/3 pi cm^3
- 25 pi cm^3
- 25/2 pi cm^3
- 25/6 pi cm^3

Answer 3

To find the volume of the sphere given the volume of the cone, let's start by recalling the formula for the volume of a cone:

\[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \]

where \( r \) is the radius and \( h \) is the height of the cone. We know that the volume of the cone is given as \( \frac{25}{3} \pi , \text{cm}^3 \).

From the volume formula, we can equate:

\[ \frac{1}{3} \pi r^2 h = \frac{25}{3} \pi \]

We can cancel \( \pi \) from both sides:

\[ \frac{1}{3} r^2 h = \frac{25}{3} \]

Multiplying both sides by 3:

\[ r^2 h = 25 \]

Now, we're told that the height of the cone (\( h \)) is equal to the diameter of the sphere, which means:

\[ h = 2r \]

Substituting \( 2r \) for \( h \) in the equation \( r^2 h = 25 \):

\[ r^2 (2r) = 25 \]

This simplifies to:

\[ 2r^3 = 25 \]

Dividing both sides by 2:

\[ r^3 = \frac{25}{2} \]

Now we can find the volume of the sphere, which is given by the formula:

\[ V_{\text{sphere}} = \frac{4}{3} \pi r^3 \]

Substituting \( r^3 \):

\[ V_{\text{sphere}} = \frac{4}{3} \pi \left(\frac{25}{2}\right) \]

This simplifies to:

\[ V_{\text{sphere}} = \frac{4 \times 25}{3 \times 2} \pi = \frac{100}{6} \pi = \frac{50}{3} \pi , \text{cm}^3 \]

Thus, the volume of the sphere is \(\boxed{\frac{50}{3} \pi , \text{cm}^3}\).