A conical cup when filled with ice - cream forms a hemispherical shape on its open end . Find the approximate volume of the ice - cream ,if the radius of the base of the cone is `3.5` cm and the vertical height of the cone is `7 cm .`

Options

A. `213 cm^(3)`
B. `190 cm^(3)`
C. `180 cm^(3)`
D. `165 cm^(3)`

Answer

To find the approximate volume of the ice cream in the conical cup, we need to calculate the volumes of both the cone and the hemisphere. The total volume of ice cream will be the sum of these two volumes.

### Step 1: Calculate the volume of the cone
The formula for the volume \( V \) of a cone is given by:
\[ V = \frac{1}{3} \pi r^2 h \]
where:
- \( r \) is the radius of the base of the cone,
- \( h \) is the height of the cone.

Given:
- Radius \( r = 3.5 \) cm,
- Height \( h = 7 \) cm.

Substituting the values into the formula:
\[ V = \frac{1}{3} \pi (3.5)^2 (7) \]

Calculating \( (3.5)^2 \):
\[ (3.5)^2 = 12.25 \]

Now substituting back:
\[ V = \frac{1}{3} \pi (12.25)(7) \]

Calculating \( 12.25 \times 7 \):
\[ 12.25 \times 7 = 85.75 \]

Now substituting this value:
\[ V = \frac{1}{3} \pi (85.75) \]

Calculating \( \frac{1}{3} \times 85.75 \):
\[ \frac{85.75}{3} \approx 28.5833 \]

So the volume of the cone is:
\[ V \approx 28.5833 \pi \text{ cm}^3 \]

Using \( \pi \approx 3.14 \):
\[ V \approx 28.5833 \times 3.14 \approx 89.78 \text{ cm}^3 \]

### Step 2: Calculate the volume of the hemisphere
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.

Since the radius of the hemisphere is the same as the radius of the cone:
- Radius \( r = 3.5 \) cm.

Substituting the value into the formula:
\[ V = \frac{2}{3} \pi (3.5)^3 \]

Calculating \( (3.5)^3 \):
\[ (3.5)^3 = 42.875 \]

Now substituting back:
\[ V = \frac{2}{3} \pi (42.875) \]

Calculating \( \frac{2}{3} \times 42.875 \):
\[ \frac{2 \times 42.875}{3} \approx 28.5833 \]

So the volume of the hemisphere is:
\[ V \approx 28.5833 \pi \text{ cm}^3 \]

Using \( \pi \approx 3.14 \):
\[ V \approx 28.5833 \times 3.14 \approx 89.78 \text{ cm}^3 \]

### Step 3: Calculate the total volume of ice cream
Now, we add the volumes of the cone and the hemisphere:
\[ \text{Total Volume} = \text{Volume of Cone} + \text{Volume of Hemisphere} \]
\[ \text{Total Volume} \approx 89.78 + 89.78 = 179.56 \text{ cm}^3 \]

### Final Answer
The approximate volume of the ice cream is:
\[ \text{Total Volume} \approx 179.56 \text{ cm}^3 \]