A cylinder a hemisphere and a cone is having same base and same height. There are of curved surfaces are in the ratio:

Options

A. `sqrt3 : sqrt3 : 1`
B. `1:sqrt3:sqrt3`
C. `sqrt2:1:sqrt3`
D. `sqrt2:sqrt2:1`

Answer

To find the ratio of the curved surface areas of a cylinder, a hemisphere, and a cone that have the same base radius \( r \) and height \( h \), we can follow these steps:

### Step 1: Write down the formulas for the curved surface areas.

1. **Curved Surface Area of a Cylinder (CSA)**:
\[ \text{CSA}_{\text{cylinder}} = 2\pi rh \]

2. **Curved Surface Area of a Hemisphere (CSA)**:
\[ \text{CSA}_{\text{hemisphere}} = 2\pi r^2 \]

3. **Curved Surface Area of a Cone (CSA)**:
\[ \text{CSA}_{\text{cone}} = \pi r l \]
where \( l \) is the slant height of the cone, which can be calculated using the Pythagorean theorem:
\[ l = \sqrt{r^2 + h^2} \]

### Step 2: Substitute the slant height into the cone's CSA formula.

Substituting \( l \) into the cone's CSA formula:
\[ \text{CSA}_{\text{cone}} = \pi r \sqrt{r^2 + h^2} \]

### Step 3: Write the ratios of the curved surface areas.

Now we can express the ratio of the curved surface areas of the cylinder, hemisphere, and cone:
\[ \text{Ratio} = \text{CSA}_{\text{cylinder}} : \text{CSA}_{\text{hemisphere}} : \text{CSA}_{\text{cone}} \]
Substituting the formulas:
\[ \text{Ratio} = 2\pi rh : 2\pi r^2 : \pi r \sqrt{r^2 + h^2} \]

### Step 4: Simplify the ratio.

We can factor out \( \pi r \) from each term:
\[ = 2h : 2r : \sqrt{r^2 + h^2} \]

### Step 5: Further simplify the ratio.

Dividing each term by \( 2r \):
\[ = \frac{2h}{2r} : \frac{2r}{2r} : \frac{\sqrt{r^2 + h^2}}{2r} \]
This simplifies to:
\[ = \frac{h}{r} : 1 : \frac{\sqrt{r^2 + h^2}}{2r} \]

### Step 6: Assign values to \( h \) and \( r \) for easier calculation.

Assuming \( h = r \) (for simplicity), we can substitute:
\[ = 1 : 1 : \frac{\sqrt{r^2 + r^2}}{2r} = 1 : 1 : \frac{\sqrt{2r^2}}{2r} = 1 : 1 : \frac{r\sqrt{2}}{2r} = 1 : 1 : \frac{\sqrt{2}}{2} \]

### Step 7: Final ratio.

Thus, the final ratio of the curved surface areas is:
\[ 1 : 1 : \frac{\sqrt{2}}{2} \]

### Conclusion

The ratio of the curved surface areas of the cylinder, hemisphere, and cone is:
\[ \text{Ratio} = 1 : 1 : \frac{\sqrt{2}}{2} \]