Suppose a sphere, cylinder, and a 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?

Question

cylinder= sphere -cone
sphere=cylinder+cone
cone=sphere-cylinder
cone=cyilinder-sphere

Answer 1

To analyze the relationship between the volumes of the sphere, cylinder, and cone, let's start by calculating their volumes using the given dimensions.

1. Volume of the sphere (V_sphere): \[ V_{\text{sphere}} = \frac{4}{3} \pi r^3 \]

2. Volume of the cylinder (V_cylinder): The height (h) of the cylinder is given to be twice the radius (r), so \( h = 2r \). \[ V_{\text{cylinder}} = \pi r^2 h = \pi r^2 (2r) = 2\pi r^3 \]

3. Volume of the cone (V_cone): The height of the cone is also \( 2r \). \[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3 \]

Now let's summarize the volumes:

• \( V_{\text{sphere}} = \frac{4}{3} \pi r^3 \)
• \( V_{\text{cylinder}} = 2 \pi r^3 \)
• \( V_{\text{cone}} = \frac{2}{3} \pi r^3 \)

Next, let's compare the volumes to determine the relationships:

1. Comparing cylinder and sphere: \[ V_{\text{cylinder}} = 2 \pi r^3 = \frac{6}{3} \pi r^3 \] Thus: \[ V_{\text{cylinder}} = V_{\text{sphere}} - V_{\text{cone}} \quad \text{(since } \frac{4}{3} - \frac{2}{3} = \frac{2}{3}) \]

2. Expanding on this: \[ 2 \pi r^3 = \frac{4}{3} \pi r^3 - \frac{2}{3} \pi r^3 \] Therefore, we indeed see that: \[ V_{\text{cylinder}} = V_{\text{sphere}} - V_{\text{cone}} \]

So, the relationship that holds true, based on the calculations, is:

cylinder = sphere - cone.

Answer 2

you sure-?

Answer 3

Let's double-check the calculations step by step to ensure the relationships among the volumes are accurate.

1. Volume of the sphere (V_sphere): \[ V_{\text{sphere}} = \frac{4}{3} \pi r^3 \]

2. Volume of the cylinder (V_cylinder): The height of the cylinder is \( h = 2r \). \[ V_{\text{cylinder}} = \pi r^2 h = \pi r^2 (2r) = 2\pi r^3 \]

3. Volume of the cone (V_cone): The height of the cone is also \( h = 2r \). \[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3 \]

Now, we summarize the volumes again:

• \( V_{\text{sphere}} = \frac{4}{3} \pi r^3 \)
• \( V_{\text{cylinder}} = 2 \pi r^3 \)
• \( V_{\text{cone}} = \frac{2}{3} \pi r^3 \)

Next, let's check the relationships among these volumes:

1. Cylinder compared to sphere and cone: We have: \[ V_{\text{sphere}} - V_{\text{cone}} = \frac{4}{3} \pi r^3 - \frac{2}{3} \pi r^3 = \frac{4 - 2}{3} \pi r^3 = \frac{2}{3} \pi r^3 \] Now, we see that: \[ V_{\text{cylinder}} = 2 \pi r^3 = \text{ not equal to } \frac{2}{3} \pi r^3 \]

2. Sphere compared to cylinder and cone: For the equation \( V_{\text{sphere}} = V_{\text{cylinder}} + V_{\text{cone}} \): \[ V_{\text{cylinder}} + V_{\text{cone}} = 2 \pi r^3 + \frac{2}{3} \pi r^3 \] To add these together, convert \( 2 \pi r^3 \) into thirds: \[ 2 \pi r^3 = \frac{6}{3} \pi r^3 \] Adding them gives: \[ \frac{6}{3} \pi r^3 + \frac{2}{3} \pi r^3 = \frac{6 + 2}{3} \pi r^3 = \frac{8}{3} \pi r^3 \] This does not equal \( \frac{4}{3} \pi r^3 \), so that relationship is incorrect.

3. Now check cone compared to cylinder: From the volumes: \[ V_{\text{cone}} = \frac{2}{3} \pi r^3, \quad V_{\text{cylinder}} - V_{\text{sphere}} = 2 \pi r^3 - \frac{4}{3} \pi r^3 = \frac{6}{3} \pi r^3 - \frac{4}{3} \pi r^3 = \frac{2}{3} \pi r^3 \] Hence, \[ V_{\text{cone}} = V_{\text{cylinder}} - V_{\text{sphere}} \] This relationship appears to be true.

4. Let's check ** c) cone=cylinder-sphere This is indeed valid.

So, the correct relationship should be:

cone = cylinder - sphere.

I apologize for the confusion in the earlier response. Thank you for your patience.