To find the volume of the whole object made up of a cone and a cylinder, we need to calculate the volume of the cylinder and then add it to the volume of the cone.
The volume \( V \) of a cone is given by the formula:
\[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \]
And the volume \( V \) of a cylinder is given by the formula:
\[ V_{\text{cylinder}} = \pi r^2 h \]
From the information given, we know:
- The volume of the cone \( V_{\text{cone}} = 50 , \text{cm}^3 \)
- The height of the cone and the cylinder are equal, which we'll call \( h \).
- The radius of both the cone and the cylinder are the same, which we will call \( r \).
Since the cone's volume is already provided, we can express the height in terms of \( r \) using the formula for the volume of the cone:
\[ 50 = \frac{1}{3} \pi r^2 h \]
To find \( h \), we can rearrange this:
\[ h = \frac{150}{\pi r^2} \]
Now we substitute this height into the formula for the cylinder's volume.
\[ V_{\text{cylinder}} = \pi r^2 h = \pi r^2 \left(\frac{150}{\pi r^2}\right) = 150 , \text{cm}^3 \]
Now we can find the total volume of the whole object, which is the sum of the volume of the cone and the volume of the cylinder:
\[ V_{\text{total}} = V_{\text{cone}} + V_{\text{cylinder}} = 50 + 150 = 200 , \text{cm}^3 \]
Thus, the volume of the whole object is 200 cm³.