To find the total volume of the combined shape (the cone on top of the cylinder), we need to calculate the volume of both the cone and the cylinder, and then add them together.
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Volume of the Cone: The volume of a cone is given by the formula: \[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \] We know the volume of the cone \( V_{\text{cone}} = 50 , \text{cm}^3 \).
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Volume of the Cylinder: The volume of a cylinder is given by the formula: \[ V_{\text{cylinder}} = \pi r^2 h \] Since the cone and the cylinder share the same radius (\( r \)) and height (\( h \)), we can express the volume of the cylinder in terms of the cone's volume.
From the cone's volume formula, we have: \[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h = 50 \] From this, we can express \( \pi r^2 h \) (the volume of the cylinder) as: \[ \pi r^2 h = 3 \times V_{\text{cone}} = 3 \times 50 = 150 , \text{cm}^3 \]
- Total Volume: Now, we can find the total volume of the object (cone + cylinder): \[ V_{\text{total}} = V_{\text{cone}} + V_{\text{cylinder}} = 50 , \text{cm}^3 + 150 , \text{cm}^3 = 200 , \text{cm}^3 \]
Thus, the volume of the whole object is \( \boxed{200 , \text{cm}^3} \).