A cone, a hemisphere and a cylinder stand on equal bases and have the same height, the height being equal to the radius of the circular base. Their total surface areas are in the ratio:

Options

A. `(sqrt(2) + 1) : 3 : 4`
B. `(sqrt(3) + 1) : 3 : 4`
C. `sqrt(2) : 3: 4`
D. `sqrt(3) : 7 : 8`

Answer

To find the ratio of the total surface areas of a cone, a hemisphere, and a cylinder that stand on equal bases and have the same height (with the height being equal to the radius of the circular base), we can follow these steps:

### Step 1: Define the variables
Let the radius of the base (r) be equal to the height (h). Therefore, we have:
- Radius (r) = Height (h) = r

### Step 2: Calculate the slant height of the cone
The slant height (l) of the cone can be calculated using the Pythagorean theorem:
\[ l = \sqrt{h^2 + r^2} \]
Since \( h = r \), we can substitute:
\[ l = \sqrt{r^2 + r^2} = \sqrt{2r^2} = r\sqrt{2} \]

### Step 3: Calculate the total surface area of the cone
The total surface area (TSA) of a cone is given by:
\[ \text{TSA}_{\text{cone}} = \pi r (r + l) \]
Substituting the value of \( l \):
\[ \text{TSA}_{\text{cone}} = \pi r \left(r + r\sqrt{2}\right) = \pi r (r(1 + \sqrt{2})) = \pi r^2 (1 + \sqrt{2}) \]

### Step 4: Calculate the total surface area of the hemisphere
The total surface area of a hemisphere is given by:
\[ \text{TSA}_{\text{hemisphere}} = 2\pi r^2 + \pi r^2 = 3\pi r^2 \]

### Step 5: Calculate the total surface area of the cylinder
The total surface area of a cylinder is given by:
\[ \text{TSA}_{\text{cylinder}} = 2\pi r h + 2\pi r^2 \]
Since \( h = r \):
\[ \text{TSA}_{\text{cylinder}} = 2\pi r^2 + 2\pi r^2 = 4\pi r^2 \]

### Step 6: Write the total surface areas in ratio form
Now we have:
- TSA of cone: \( \pi r^2 (1 + \sqrt{2}) \)
- TSA of hemisphere: \( 3\pi r^2 \)
- TSA of cylinder: \( 4\pi r^2 \)

The ratio of the total surface areas is:
\[ \text{Ratio} = \pi r^2 (1 + \sqrt{2}) : 3\pi r^2 : 4\pi r^2 \]
Cancelling \( \pi r^2 \) from all terms, we get:
\[ (1 + \sqrt{2}) : 3 : 4 \]

### Final Answer
The total surface areas of the cone, hemisphere, and cylinder are in the ratio:
\[ (1 + \sqrt{2}) : 3 : 4 \]

---