The volume of a cone is c * m ^ 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? (1 point)
The formula for the volume of a cone is given by:
\[ V_{cone} = \frac{1}{3} \pi r^2 h \]
where \( r \) is the radius of the base of the cone and \( h \) is its height. We are given that the volume of the cone is \( c \cdot m^3 \), although normally \( c \) would likely represent a constant factor. However, we'll just use \( c \cdot m^3 \) as the given volume in your question.
For the sphere with the same radius as the cone, the formula for the volume of a sphere is given by:
\[ V_{sphere} = \frac{4}{3} \pi r^3 \]
Since we are told the height of the cone (\( h \)) is equal to the diameter of the sphere (\( 2r \)), we can write:
\[ h = 2r \]
First, we solve for the radius \( r \) in terms of the cone's volume \( c \cdot m^3 \) using the cone's volume formula:
\[ c \cdot m^3 = \frac{1}{3} \pi r^2 h \]
\[ c \cdot m^3 = \frac{1}{3} \pi r^2 (2r) \]
\[ c \cdot m^3 = \frac{2}{3} \pi r^3 \]
Now, let's solve for \( r^3 \):
\[ r^3 = \frac{3c \cdot m^3}{2\pi} \]
Next, we plug this expression for \( r^3 \) into the sphere's volume formula:
\[ V_{sphere} = \frac{4}{3} \pi r^3 \]
\[ V_{sphere} = \frac{4}{3} \pi \left( \frac{3c \cdot m^3}{2\pi} \right) \]
Simplify the expression:
\[ V_{sphere} = 2c \cdot m^3 \]
Therefore, the volume of the sphere, given that its radius is equal to the cone's and the height of the cone is equal to the sphere's diameter, is \( 2c \cdot m^3 \).