To find the volume of the sphere given the volume of the cone and the relationships between the dimensions, we start with the provided volume of the cone, which is:
\[ V_{\text{cone}} = \frac{25}{3} \pi \text{ cm}^3 \]
The volume of a cone can be calculated using the formula:
\[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \]
where \( r \) is the radius of the cone and \( h \) is its height.
Let’s denote the radius of the cone as \( r \) and the height as \( h \). Using the formula for the volume of the cone, we can set it equal to the known volume:
\[ \frac{1}{3} \pi r^2 h = \frac{25}{3} \pi \]
Simplifying this (by multiplying both sides by 3 and dividing by \( \pi \)) gives:
\[ r^2 h = 25 \]
We also know from the problem statement:
- The radius \( r \) of the cone is the same as the radius of the sphere.
- The height \( h \) of the cone is equal to the diameter of the sphere, which is \( 2r \).
Thus, we can substitute \( h \) with \( 2r \) in the equation:
\[ r^2 (2r) = 25 \]
This simplifies to:
\[ 2r^3 = 25 \]
Now we can solve for \( r \):
\[ r^3 = \frac{25}{2} \]
Next, we calculate the radius \( r \):
\[ r = \left(\frac{25}{2}\right)^{1/3} \]
Now we need to find the volume of the sphere, which can be calculated using the formula:
\[ V_{\text{sphere}} = \frac{4}{3} \pi r^3 \]
We already found that \( r^3 = \frac{25}{2} \), so we can substitute that into the volume formula for the sphere:
\[ V_{\text{sphere}} = \frac{4}{3} \pi \left(\frac{25}{2}\right) \]
Calculating this gives:
\[ V_{\text{sphere}} = \frac{4 \cdot 25}{3 \cdot 2} \pi = \frac{100}{6} \pi = \frac{50}{3} \pi \text{ cm}^3 \]
Thus, the volume of the sphere is:
\[ \frac{50}{3} \pi \text{ cm}^3 \]
Therefore, the correct answer is:
\[ \frac{50}{3} \pi \text{ cm}^3 \]
1 answer