To find the volume of the whole object, which consists of both the cone and the cylinder, we first need to calculate the volume of the cylinder.
We are given that the volume of the cone \( V_{cone} \) is 50 cm³. The formula for the volume of a cone is:
\[ V_{cone} = \frac{1}{3} \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height.
Since we know that the cone and cylinder have the same radius and height, we can use the same \( r \) and \( h \) for the cylinder. The formula for the volume of a cylinder is:
\[ V_{cylinder} = \pi r^2 h \]
Next, let’s calculate the height of the cone, which we can derive from the volume of the cone formula. Given \( V_{cone} = 50 \) cm³, we have:
\[ 50 = \frac{1}{3} \pi r^2 h \implies r^2 h = \frac{150}{\pi} \]
Now, we can substitute this into the volume of the cylinder:
\[ V_{cylinder} = \pi r^2 h = \pi \left(\frac{150}{\pi h}\right) = 150 \]
The total volume of the object (cone + cylinder) is:
\[ V_{total} = V_{cone} + V_{cylinder} = 50 + 150 = 200 \text{ cm³} \]
Thus, the volume of the whole object is:
\[ \boxed{200} \text{ cm³} \]
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