The volume of a cone is 20/3 * pi*c * m ^ 3 What is the volume of a sphere it its radius is the same as the cone's and the height of the cone is equal to the sphere's diameter? (1 point)
25c * m ^ 3
26/3 * c * m ^ 3
50/3 * pi*c * m ^ 3
25c * m ^ 3
To solve this problem, let's start by understanding the formula for the volume of a cone and a sphere.
The volume V of a cone is given by:
\[ V_{cone} = \frac{1}{3}\pi r^2 h \]
where \( r \) is the radius and \( h \) is the height of the cone.
The volume V of a sphere is given by:
\[ V_{sphere} = \frac{4}{3}\pi r^3 \]
where \( r \) is the radius of the sphere.
We're given that the volume of the cone is \( \frac{20}{3}\pi c \ m^3 \), and we need to find the volume of a sphere whose radius is the same as the cone's, with the height of the cone equal to the sphere's diameter.
Since the height of the cone is equal to the sphere's diameter, we can deduce that:
\[ h = 2r \]
Given that the volume of the cone is \( \frac{20}{3}\pi c \), we can set that equal to the cone's volume formula and solve for \( r \):
\[ \frac{20}{3}\pi c = \frac{1}{3}\pi r^2 h \]
\[ \frac{20}{3}\pi c = \frac{1}{3}\pi r^2 (2r) \]
\[ 20c = 2r^3 \]
\[ r^3 = \frac{20c}{2} \]
\[ r^3 = 10c \]
Now that we have \( r^3 \), we can substitute it into the sphere's volume formula:
\[ V_{sphere} = \frac{4}{3}\pi r^3 \]
\[ V_{sphere} = \frac{4}{3}\pi (10c) \]
\[ V_{sphere} = \frac{40}{3}\pi c \]
Thus, the volume of the sphere is \( \frac{40}{3}\pi c \ m^3 \), which is not listed exactly as it was presented in the options. However, taking into account the pi symbol, the closest matching option is:
50/3 * pi*c * m ^ 3
This suggests that there might be a typo in the available options. The correct volume, including the pi symbol, would be:
\[ \frac{40}{3}\pi c \ m^3 \]