A solid wooden toy is in the shape of a right circular cone mounted on a hemisphere. If the radius of the hemisphere is 4.2 cm and the total height of the toy is 10.2 cm, find the volume of the wooden toy (nearly).

Options

A. `104 cm^(3)`
B. `162 cm^(3)`
C. `427 cm^(3)`
D. `266 cm^(3)`

Answer

To find the volume of the solid wooden toy shaped like a right circular cone mounted on a hemisphere, we will follow these steps:

### Step 1: Identify the given values
- Radius of the hemisphere (r) = 4.2 cm
- Total height of the toy (H) = 10.2 cm

### Step 2: Calculate the height of the cone
The height of the cone (h) can be found by subtracting the radius of the hemisphere from the total height of the toy.
\[ h = H - r = 10.2 \, \text{cm} - 4.2 \, \text{cm} = 6 \, \text{cm} \]

### Step 3: Calculate the volume of the hemisphere
The formula for the volume of a hemisphere is given by:
\[ V_{\text{hemisphere}} = \frac{2}{3} \pi r^3 \]
Substituting the value of the radius:
\[ V_{\text{hemisphere}} = \frac{2}{3} \times \frac{22}{7} \times (4.2)^3 \]
Calculating \( (4.2)^3 \):
\[ (4.2)^3 = 74.088 \]
Now substituting this value back into the volume formula:
\[ V_{\text{hemisphere}} = \frac{2}{3} \times \frac{22}{7} \times 74.088 \approx 220.8 \, \text{cm}^3 \]

### Step 4: Calculate the volume of the cone
The formula for the volume of a cone is given by:
\[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \]
Substituting the values of the radius and height:
\[ V_{\text{cone}} = \frac{1}{3} \times \frac{22}{7} \times (4.2)^2 \times 6 \]
Calculating \( (4.2)^2 \):
\[ (4.2)^2 = 17.64 \]
Now substituting this value back into the volume formula:
\[ V_{\text{cone}} = \frac{1}{3} \times \frac{22}{7} \times 17.64 \times 6 \]
Calculating:
\[ V_{\text{cone}} \approx \frac{1}{3} \times \frac{22}{7} \times 105.84 \approx 79.2 \, \text{cm}^3 \]

### Step 5: Calculate the total volume of the wooden toy
Now, we can find the total volume of the wooden toy by adding the volumes of the hemisphere and the cone:
\[ V_{\text{total}} = V_{\text{hemisphere}} + V_{\text{cone}} \approx 220.8 \, \text{cm}^3 + 79.2 \, \text{cm}^3 = 300 \, \text{cm}^3 \]

### Final Answer
The volume of the wooden toy is approximately \( 300 \, \text{cm}^3 \).

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