A buoy is made in the form of hemisphere surmounted by a right cone whose circular base coincides with the plane surface of hemisphere. The radius of the base of the cone is 3.5 metres and its volume is two-third of the hemisphere. Calculate the height of the cone and the surface area of the buoy, correct to two places of decimal.

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Identify the given values
- Radius of the base of the cone and hemisphere, \( r = 3.5 \) m.
- Volume of the cone is \( \frac{2}{3} \) of the volume of the hemisphere.

### Step 2: 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 \( r \):
\[ V_{\text{hemisphere}} = \frac{2}{3} \pi (3.5)^3 \]
Calculating \( (3.5)^3 \):
\[ (3.5)^3 = 42.875 \]
Now substituting this value back:
\[ V_{\text{hemisphere}} = \frac{2}{3} \times \pi \times 42.875 \]
Using \( \pi \approx \frac{22}{7} \):
\[ V_{\text{hemisphere}} = \frac{2}{3} \times \frac{22}{7} \times 42.875 \]
Calculating further:
\[ V_{\text{hemisphere}} = \frac{2 \times 22 \times 42.875}{3 \times 7} = \frac{1879.25}{21} \approx 89.95 \text{ m}^3 \]

### Step 3: Calculate the volume of the cone
The volume of the cone is given as:
\[ V_{\text{cone}} = \frac{2}{3} V_{\text{hemisphere}} = \frac{2}{3} \times 89.95 \approx 59.97 \text{ m}^3 \]

### Step 4: Use the volume of the cone to find the height
The formula for the volume of a cone is:
\[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \]
Substituting the known values:
\[ 59.97 = \frac{1}{3} \times \frac{22}{7} \times (3.5)^2 \times h \]
Calculating \( (3.5)^2 \):
\[ (3.5)^2 = 12.25 \]
Now substituting this value:
\[ 59.97 = \frac{1}{3} \times \frac{22}{7} \times 12.25 \times h \]
Calculating the constant:
\[ \frac{22 \times 12.25}{21} = \frac{271.5}{21} \approx 12.93 \]
Now we have:
\[ 59.97 = 12.93h \]
Solving for \( h \):
\[ h = \frac{59.97}{12.93} \approx 4.64 \text{ m} \]

### Step 5: Calculate the slant height of the cone
Using the Pythagorean theorem:
\[ l = \sqrt{r^2 + h^2} \]
Substituting the values:
\[ l = \sqrt{(3.5)^2 + (4.64)^2} = \sqrt{12.25 + 21.53} = \sqrt{33.78} \approx 5.81 \text{ m} \]

### Step 6: Calculate the surface area of the buoy
The surface area of the buoy is the sum of the curved surface area of the cone and the curved surface area of the hemisphere:
\[ \text{Surface Area} = \text{CSA of cone} + \text{CSA of hemisphere} \]
Curved surface area of the cone:
\[ \text{CSA}_{\text{cone}} = \pi r l = \frac{22}{7} \times 3.5 \times 5.81 \]
Calculating:
\[ \text{CSA}_{\text{cone}} \approx 22 \times 0.5 \times 5.81 \approx 64.05 \text{ m}^2 \]
Curved surface area of the hemisphere:
\[ \text{CSA}_{\text{hemisphere}} = 2 \pi r^2 = 2 \times \frac{22}{7} \times (3.5)^2 \]
Calculating:
\[ \text{CSA}_{\text{hemisphere}} \approx 2 \times \frac{22}{7} \times 12.25 \approx 50.57 \text{ m}^2 \]
Total surface area:
\[ \text{Surface Area} = 64.05 + 50.57 \approx 114.62 \text{ m}^2 \]

### Final Results
- Height of the cone: \( \approx 4.64 \text{ m} \)
- Surface area of the buoy: \( \approx 114.62 \text{ m}^2 \)