A toy is in the shape of a cone mounted on a hemisphere, both of radius 7 cm. The total height of the toy is 21 cm. Find the volume of the toy. (Use π = 22/7.)

Step 1: Identify Given Values and Calculate Cone Height

We are given that the radius of both the cone and the hemisphere is $R = 7 \text{ cm}$. The total height of the toy is $H = 21 \text{ cm}$. Since the hemisphere has a radius of $7 \text{ cm}$, its height is also $7 \text{ cm}$. To find the height of the cone ($h$), we subtract the height of the hemisphere from the total height of the toy.

Step 2: Calculate Height of the Cone

Substituting the given values, the height of the cone is $21 \text{ cm} - 7 \text{ cm} = 14 \text{ cm}$.

Step 3: Calculate Volume of Hemisphere

The formula for the volume of a hemisphere is $\frac{2}{3} \pi R^3$. We will use this formula to calculate the volume of the hemispherical part of the toy.

Step 4: Substitute Values for Hemisphere Volume

Substitute $R = 7 \text{ cm}$ and $\pi = \frac{22}{7}$ into the hemisphere volume formula. After calculation, the volume of the hemisphere is $\frac{2156}{3} \text{ cm}^3$.

Step 5: Calculate Volume of Cone

The formula for the volume of a cone is $\frac{1}{3} \pi R^2 h$. We will use this formula to calculate the volume of the conical part of the toy.

Step 6: Substitute Values for Cone Volume

Substitute $R = 7 \text{ cm}$, $h = 14 \text{ cm}$, and $\pi = \frac{22}{7}$ into the cone volume formula. After calculation, the volume of the cone is $\frac{2156}{3} \text{ cm}^3$.

Step 7: Calculate Total Volume of Toy

The total volume of the toy is the sum of the volume of the hemisphere and the volume of the cone. Adding the two volumes, we get $\frac{2156}{3} + \frac{2156}{3} = \frac{4312}{3} \text{ cm}^3$, which is approximately $1437.33 \text{ cm}^3$.

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