To find the total volume of the object made of a cone on top of a cylinder, we first need to find the volume of the cylinder and then add it to the volume of the cone.
Given:
- Volume of the cone \( V_{cone} = \frac{1}{3} \pi r^2 h \)
- Volume of the cylinder \( V_{cylinder} = \pi r^2 h \)
According to the information provided:
- The volume of the cone is \( 50 , \text{cm}^3 \).
- The cone and the cylinder have the same radius \( r \) and height \( h \).
Let’s denote the volume of the cone as: \[ V_{cone} = 50 , \text{cm}^3 \]
The volume of the cylinder can be expressed in terms of the volume of the cone as follows: \[ V_{cylinder} = 3 V_{cone} \] because the volume of the cone is a third of the volume of the cylinder when they have the same radius and height.
Thus, we can calculate the volume of the cylinder: \[ V_{cylinder} = 3 \times 50 , \text{cm}^3 = 150 , \text{cm}^3 \]
Now, we find the total volume of the whole object (cone + cylinder): \[ V_{total} = V_{cone} + V_{cylinder} = 50 , \text{cm}^3 + 150 , \text{cm}^3 = 200 , \text{cm}^3 \]
Therefore, the volume of the whole object is: \[ \boxed{200 , \text{cm}^3} \]
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