To find the volume of the solid, we first need to calculate the volume of both the cone and the hemisphere separately, and then add them together.
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Volume of the Hemisphere:
The volume \( V_h \) of a hemisphere is given by: \[ V_h = \frac{2}{3} \pi r^3 \] where \( r \) is the radius of the hemisphere. In this case, the radius \( r = 1 \) cm. Substituting this value into the formula: \[ V_h = \frac{2}{3} \pi (1)^3 = \frac{2}{3} \pi \text{ cm}^3 \]
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Volume of the Cone:
The volume \( V_c \) of a cone is given by: \[ V_c = \frac{1}{3} \pi r^2 h \] where \( r \) is the radius of the base of the cone and \( h \) is the height of the cone. In this case, the radius \( r = 1 \) cm, and the height \( h = 1 \) cm (since the height of the cone is equal to its radius). Substituting these values into the formula: \[ V_c = \frac{1}{3} \pi (1)^2 (1) = \frac{1}{3} \pi \text{ cm}^3 \]
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Total Volume of the Solid:
The total volume \( V_{total} \) of the solid is the sum of the volume of the hemisphere and the volume of the cone: \[ V_{total} = V_h + V_c \] Now we can substitute the volumes we calculated: \[ V_{total} = \frac{2}{3} \pi + \frac{1}{3} \pi \]
Combining these fractions: \[ V_{total} = \left(\frac{2}{3} + \frac{1}{3}\right) \pi = \frac{3}{3} \pi = \pi \text{ cm}^3 \]
Thus, the total volume of the solid in terms of \( \pi \) is: \[ \boxed{\pi} \text{ cm}^3 \]