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? (1 point)
cone = sphere - c * yhn * det
cyhin * der = sphere - cone
sphere = cylin * der + cone
cone = cyhn * der - sphere
First, let's write down the formulas for the volume of each of these shapes:
For a sphere of radius \( r \):
\[ V_{sphere} = \frac{4}{3}\pi r^3 \]
For a cylinder with radius \( r \) and height \( h \) (which is \( 2r \) in this case):
\[ V_{cylinder} = \pi r^2 h = \pi r^2 (2r) = 2\pi r^3 \]
For a cone with the same radius and height as the cylinder:
\[ V_{cone} = \frac{1}{3}\pi r^2 h = \frac{1}{3}\pi r^2 (2r) = \frac{2}{3}\pi r^3 \]
We are asked to find the relationship between these formulas. Let's inspect the volumes in relation to each other:
Given the cylinder's height is twice the radius (\( h = 2r \)), the volume of the cylinder is:
\[ V_{cylinder} = 2\pi r^3 \]
The sphere's volume is:
\[ V_{sphere} = \frac{4}{3}\pi r^3 \]
And the cone's volume is a third of the cylinder's:
\[ V_{cone} = \frac{2}{3}\pi r^3 \]
From this, we can see the following relationship:
\[ V_{cylinder} = 2 \cdot V_{cone} \]
\[ V_{cylinder} = \frac{2}{3}\pi r^3 + \frac{2}{3}\pi r^3 = \frac{2}{3}\pi r^3 + \frac{2}{3}\pi r^3 = 2 \cdot \frac{2}{3}\pi r^3 = V_{cylinder} \]
The sphere's volume can be expressed in terms of the cone's and cylinder's volumes:
\[ V_{sphere} = V_{cylinder} - V_{cone} \]
\[ V_{sphere} = 2\pi r^3 - \frac{2}{3}\pi r^3 \]
\[ V_{sphere} = \frac{6}{3}\pi r^3 - \frac{2}{3}\pi r^3 \]
\[ V_{sphere} = \frac{4}{3}\pi r^3 \]
Indeed, we have:
\[ \frac{4}{3}\pi r^3 = 2\pi r^3 - \frac{2}{3}\pi r^3 \]
Therefore, the relationship that is true is:
\[ sphere = cylin \cdot der + cone \]
Note: The placeholders "cylin," "der," and "cone" correspond to "cylinder," "derivative" (likely a typo for a part of 'cylinder' or just 'der' as a connector), and "cone" respectively. The correct relationship simplifies the equation by replacing these placeholders with the appropriate mathematical terms. The original options provided seem to include these placeholders, which don't directly correspond to mathematical entities or operations.