A solid metallic toy has a hemispherical base with radius `3.5` cm surmounted by a cone. If the height of the cone is the same as the radius of its base, then the volume of metal used in `cm^3` is : (Take `pi = 22/7`)

Options

A. `89 2/3 cm^3`
B. `144 3/4 cm^3`
C. `134 3/4 cm^3`
D. `179 2/3 cm^3`

Answer

To find the volume of the solid metallic toy, we need to calculate the volume of both the hemispherical base and the cone that is surmounted on it.

### Step 1: Calculate the volume of the hemispherical base.

The formula for the volume \( V \) of a hemisphere is given by:

\[ V = \frac{2}{3} \pi r^3 \]

Where \( r \) is the radius of the hemisphere. Given that the radius \( r = 3.5 \) cm, we can substitute this value into the formula.

\[ V = \frac{2}{3} \times \frac{22}{7} \times (3.5)^3 \]

### Step 2: Calculate \( (3.5)^3 \).

First, we calculate \( (3.5)^3 \):

\[ (3.5)^3 = 3.5 \times 3.5 \times 3.5 = 12.25 \times 3.5 = 42.875 \]

### Step 3: Substitute \( (3.5)^3 \) back into the volume formula.

Now substitute \( 42.875 \) back into the volume formula:

\[ V = \frac{2}{3} \times \frac{22}{7} \times 42.875 \]

### Step 4: Calculate the volume of the cone.

The cone has a radius equal to the radius of the hemisphere, which is \( 3.5 \) cm, and its height is the same as the radius of its base, which is also \( 3.5 \) cm. The formula for the volume \( V \) of a cone is:

\[ V = \frac{1}{3} \pi r^2 h \]

Substituting \( r = 3.5 \) cm and \( h = 3.5 \) cm into the formula:

\[ V = \frac{1}{3} \times \frac{22}{7} \times (3.5)^2 \times 3.5 \]

### Step 5: Calculate \( (3.5)^2 \).

First, calculate \( (3.5)^2 \):

\[ (3.5)^2 = 12.25 \]

### Step 6: Substitute \( (3.5)^2 \) back into the volume formula for the cone.

Now substitute \( 12.25 \) back into the volume formula:

\[ V = \frac{1}{3} \times \frac{22}{7} \times 12.25 \times 3.5 \]

### Step 7: Combine the volumes.

Now we need to add the volumes of the hemisphere and the cone together to find the total volume of the metallic toy.

1. Calculate the volume of the hemisphere.
2. Calculate the volume of the cone.
3. Add both volumes together.

### Final Calculation:

1. Volume of the hemisphere:
\[ V_{hemisphere} = \frac{2}{3} \times \frac{22}{7} \times 42.875 \approx 100.5 \, cm^3 \]

2. Volume of the cone:
\[ V_{cone} = \frac{1}{3} \times \frac{22}{7} \times 12.25 \times 3.5 \approx 28.5 \, cm^3 \]

3. Total volume:
\[ V_{total} = V_{hemisphere} + V_{cone} \approx 100.5 + 28.5 = 129 \, cm^3 \]

### Conclusion:

The volume of metal used in the toy is approximately \( 129 \, cm^3 \).

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