Suppose a sphere and a cone share the same radius and the height of the cone is twice the radius. Which ratio describes the relationship between the formulas for the volume of a cone to the volume of a sphere?

Question

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Answer 1

To find the ratio of the volume of a cone to the volume of a sphere when they share the same radius \( r \) and the height of the cone is twice the radius, we first need to calculate the volumes of both shapes.

1. Volume of a sphere: The formula for the volume \( V_s \) of a sphere is given by: \[ V_s = \frac{4}{3} \pi r^3 \]

2. Volume of a cone: The formula for the volume \( V_c \) of a cone is given by: \[ V_c = \frac{1}{3} \pi r^2 h \] In our case, the height \( h \) of the cone is twice the radius: \[ h = 2r \] Substituting this into the volume formula for the cone gives: \[ V_c = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3 \]

Now we can find the ratio of the volume of the cone to the volume of the sphere: \[ \text{Ratio} = \frac{V_c}{V_s} = \frac{\frac{2}{3} \pi r^3}{\frac{4}{3} \pi r^3} \]

The \( \pi r^3 \) terms will cancel out: \[ \text{Ratio} = \frac{\frac{2}{3}}{\frac{4}{3}} = \frac{2}{4} = \frac{1}{2} \]

Thus, the ratio that describes the relationship between the formulas for the volume of a cone to the volume of a sphere is:

\[ \boxed{1:2} \]

Answer 2

Suppose a sphere, cylinder, and cone all share the same radius. The cylinder and cone also share the same height, which is twice the radius. Which of the following is true about the relationship between the volume formulas for the sphere, cylinder, and cone?

cone = sphere - cylinder

sphere = cylinder + cone

cylinder = sphere - cone

cone = cylinder - sphere

Answer 3

Let's find the volumes of the sphere, cylinder, and cone given that they share the same radius \( r \) and that the height \( h \) of the cylinder and cone is twice the radius, \( h = 2r \).

1. Volume of a sphere: The formula for the volume \( V_s \) of a sphere is: \[ V_s = \frac{4}{3} \pi r^3 \]

2. Volume of a cylinder: The formula for the volume \( V_cyl \) of a cylinder is: \[ V_cyl = \pi r^2 h \] Substituting \( h = 2r \) into the formula gives: \[ V_cyl = \pi r^2 (2r) = 2 \pi r^3 \]

3. Volume of a cone: The formula for the volume \( V_{cone} \) of a cone is: \[ V_{cone} = \frac{1}{3} \pi r^2 h \] Again substituting \( h = 2r \): \[ V_{cone} = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3 \]

Now we have the volumes:

• Volume of the sphere: \( V_s = \frac{4}{3} \pi r^3 \)
• Volume of the cylinder: \( V_{cyl} = 2 \pi r^3 \)
• Volume of the cone: \( V_{cone} = \frac{2}{3} \pi r^3 \)

Now we can check each of the proposed relationships:

1. cone = sphere - cylinder: Substituting the volumes: \[ \frac{2}{3} \pi r^3 = \frac{4}{3} \pi r^3 - 2 \pi r^3 \] Simplifying: \[ \frac{2}{3} \pi r^3 = \frac{4}{3} \pi r^3 - \frac{6}{3} \pi r^3 = -\frac{2}{3} \pi r^3 \quad (\text{False}) \]

2. sphere = cylinder + cone: Substituting the volumes: \[ \frac{4}{3} \pi r^3 = 2 \pi r^3 + \frac{2}{3} \pi r^3 \] Simplifying: \[ \frac{4}{3} \pi r^3 = \frac{6}{3} \pi r^3 + \frac{2}{3} \pi r^3 = \frac{8}{3} \pi r^3 \quad (\text{False}) \]

3. cylinder = sphere - cone: Substituting the volumes: \[ 2 \pi r^3 = \frac{4}{3} \pi r^3 - \frac{2}{3} \pi r^3 \] Simplifying: \[ 2 \pi r^3 = \frac{2}{3} \pi r^3 \quad (\text{False}) \]

4. cone = cylinder - sphere: Substituting the volumes: \[ \frac{2}{3} \pi r^3 = 2 \pi r^3 - \frac{4}{3} \pi r^3 \] Simplifying: \[ \frac{2}{3} \pi r^3 = \frac{6}{3} \pi r^3 - \frac{4}{3} \pi r^3 = \frac{2}{3} \pi r^3 \quad (\text{True}) \]

Thus, the true relationship is:

\[ \text{cone} = \text{cylinder} - \text{sphere} \]

\[ \boxed{\text{cone} = \text{cylinder} - \text{sphere}} \]

Answer 4

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 cone's and the height of the cone is equal to the sphere's diameter

25/6 pi cm^3

25 pi cm^3

50/3 pi cm^3

25/2 pi cm^3

Answer 5

To find the volume of the sphere with the same radius as the cone, we need to determine the radius of the cone first from the given volume. The volume \( V \) of a cone is given by the formula:

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

We know from the problem that the volume of the cone is:

\[ \frac{25}{3} \pi , \text{cm}^3 \]

From this, we can set up the equation:

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

To simplify, we can cancel out \( \pi \) from both sides (assuming \( \pi \neq 0 \)):

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

Multiplying both sides by 3:

\[ r^2 h = 25 \]

Step 1: Determine \( h \)

According to the problem, the height \( h \) of the cone is equal to the diameter of the sphere. The diameter of the sphere is \( 2r \), thus:

\[ h = 2r \]

Substituting \( h \) into the volume equation:

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

This simplifies to:

\[ 2r^3 = 25 \]

Step 2: Solve for \( r^3 \)

Rearranging gives:

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

Step 3: Volume of the sphere

The volume \( V_{sphere} \) of a sphere is given by the formula:

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

Substituting \( r^3 = \frac{25}{2} \):

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

Calculating this gives:

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

Thus, the volume of the sphere is:

\[ \boxed{\frac{50}{3} \pi , \text{cm}^3} \]