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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 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?
54 cm^3
18 cm^3
12 cm^3
24 cm^3

3 answers

To solve the problem, we'll start with the formulas for the volumes of a cylinder and a sphere.

  1. Volume of a Cylinder: \[ V_{\text{cylinder}} = \pi r^2 h \] where \( r \) is the radius and \( h \) is the height.

  2. Volume of a Sphere: \[ V_{\text{sphere}} = \frac{4}{3} \pi r^3 \]

According to the given information:

  • The volume of the cylinder is \( 36 , \text{cm}^3 \).
  • The height of the cylinder equals the diameter of the sphere, which can be expressed as \( h = 2r \).

Substituting \( h \) into the cylinder's volume equation: \[ V_{\text{cylinder}} = \pi r^2 (2r) = 2\pi r^3 \] We know that: \[ 2\pi r^3 = 36 \] Now, we solve for \( r^3 \): \[ \pi r^3 = \frac{36}{2} = 18 \] \[ r^3 = \frac{18}{\pi} \]

Next, we can find the volume of the sphere using the radius \( r \): \[ V_{\text{sphere}} = \frac{4}{3} \pi r^3 \] Substituting \( r^3 \) from our previous calculation: \[ V_{\text{sphere}} = \frac{4}{3} \pi \left( \frac{18}{\pi} \right) \]

Now, simplify this: \[ V_{\text{sphere}} = \frac{4}{3} \cdot 18 = \frac{72}{3} = 24 , \text{cm}^3 \]

Thus, the volume of the sphere is 24 cm³.

So, the correct answer is: \[ \boxed{24 , \text{cm}^3} \]

just wanna make sure are you correct-?

Let's go through the calculations step-by-step to ensure everything is correct.

  1. Volume of Cylinder:

    • The volume of the cylinder is given by: \[ V_{\text{cylinder}} = \pi r^2 h \]
    • We know \( V_{\text{cylinder}} = 36 , \text{cm}^3 \).

  2. Height of the Cylinder:

    • The height \( h \) of the cylinder is equal to the diameter of the sphere, which is \( h = 2r \).

  3. Substituting Height into the Cylinder Volume Formula:

    • Substitute \( h \) in the volume equation: \[ 36 = \pi r^2 (2r) \] \[ 36 = 2\pi r^3 \]

  4. Solving for \( r^3 \):

    • Rearranging gives: \[ \pi r^3 = \frac{36}{2} = 18 \] \[ r^3 = \frac{18}{\pi} \]

  5. Volume of the Sphere:

    • Now, use the radius \( r \) to find the volume of the sphere: \[ V_{\text{sphere}} = \frac{4}{3} \pi r^3 \]
    • Substitute \( r^3 \) into the sphere's volume: \[ V_{\text{sphere}} = \frac{4}{3} \pi \left( \frac{18}{\pi} \right) \]
    • The \( \pi \) cancels out: \[ V_{\text{sphere}} = \frac{4}{3} \cdot 18 = \frac{72}{3} = 24 , \text{cm}^3 \]

Therefore, upon reviewing, every step has been confirmed to be correct, and the final volume of the sphere is indeed \( 24 , \text{cm}^3 \).

The answer is: \[ \boxed{24 , \text{cm}^3} \]