Suppose a spear a cylinder and Cone I'll 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 spear cylinder and tone

Answer 1

Let's start by writing out the formulas for the volumes of a sphere, a cylinder, and a cone, and then compare them given that the cylinder and cone have the same height (which is twice the radius) as well as the same radius as the sphere.

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

2. Volume of a cylinder (V_cylinder):
\[ V_{\text{cylinder}} = \pi r^2 h \]
Given h = 2r (the height is twice the radius),
\[ V_{\text{cylinder}} = \pi r^2 \cdot 2r = 2\pi r^3 \]

3. Volume of a cone (V_cone):
\[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \]
Given h = 2r (the height is twice the radius),
\[ V_{\text{cone}} = \frac{1}{3} \pi r^2 \cdot 2r = \frac{2}{3} \pi r^3 \]

Now that we have the formulas based on the same radius and height for the cylinder and cone, let's compare their volumes:

- The volume of the sphere is proportional to the cube of the radius, multiplied by \( \frac{4}{3} \pi \).
- The volume of the cylinder, given its height is twice its radius, is \( 2\pi r^3 \), which is 3 times the volume of the cone \((3 \times \frac{2}{3} \pi r^3 = 2\pi r^3)\).
- The volume of the cone is a third of the volume of the cylinder due to the \(\frac{1}{3}\) factor in its formula.

Using this relationship, you can see the volumes of the cylinder, cone, and sphere with the same radius and with the height of the cylinder and cone being equal to twice the radius, will be in the ratio:

\[ V_{\text{sphere}} : V_{\text{cylinder}} : V_{\text{cone}} = \frac{4}{3} \pi r^3 : 2\pi r^3 : \frac{2}{3} \pi r^3 \]

Reducing each term by \( \pi r^3 \):

\[ = \frac{4}{3} : 2 : \frac{2}{3} \]

We could also express these ratios with common denominators to make them easier to compare, resulting in (assuming \( r \neq 0 \)):

\[ = 4:6:2 \]

When simplified:

\[ = 2:3:1 \]

From this comparison, we can conclude:
- The volume of the sphere is 2 times the volume of the cone.
- The volume of the cylinder is 3 times the volume of the cone.
- The volume of the cylinder is 1.5 times the volume of the sphere.