A hemishpere and a cone have equal base. If their heights are also equal, the ratio of their curved surface will be :

Options

A. `1:sqrt(2)`
B. `sqrt(2) :1`
C. `1:2`
D. `2:1`

Answer

To find the ratio of the curved surface areas of a hemisphere and a cone with equal bases and equal heights, we can follow these steps:

### Step 1: Define the Variables
Let the radius of both the hemisphere and the cone be \( R \). Since they have equal bases, their radius is the same.

### Step 2: Determine the Height
The height of the hemisphere is equal to its radius, so the height \( h \) of the hemisphere is:
\[ h = R \]
For the cone, the height is also given to be equal to the radius, so:
\[ h' = R \]

### Step 3: Calculate the Curved Surface Area of the Hemisphere
The formula for the curved surface area (CSA) of a hemisphere is:
\[ \text{CSA}_{\text{hemisphere}} = 2\pi R^2 \]

### Step 4: Calculate the Curved Surface Area of the Cone
The formula for the curved surface area of a cone is:
\[ \text{CSA}_{\text{cone}} = \pi R l \]
where \( l \) is the slant height of the cone.

### Step 5: Calculate the Slant Height of the Cone
The slant height \( l \) can be calculated using the Pythagorean theorem:
\[ l = \sqrt{R^2 + h'^2} \]
Since \( h' = R \), we have:
\[ l = \sqrt{R^2 + R^2} = \sqrt{2R^2} = R\sqrt{2} \]

### Step 6: Substitute the Slant Height into the Cone's CSA
Now substitute \( l \) into the formula for the cone's curved surface area:
\[ \text{CSA}_{\text{cone}} = \pi R (R\sqrt{2}) = \pi R^2 \sqrt{2} \]

### Step 7: Find the Ratio of the Curved Surface Areas
Now we can find the ratio of the curved surface areas of the hemisphere to the cone:
\[ \text{Ratio} = \frac{\text{CSA}_{\text{hemisphere}}}{\text{CSA}_{\text{cone}}} = \frac{2\pi R^2}{\pi R^2 \sqrt{2}} \]

### Step 8: Simplify the Ratio
Cancel out \( \pi R^2 \) from the numerator and denominator:
\[ \text{Ratio} = \frac{2}{\sqrt{2}} = \frac{2\sqrt{2}}{2} = \sqrt{2} \]

### Conclusion
Thus, the ratio of the curved surface areas of the hemisphere to the cone is:
\[ \text{Ratio} = \sqrt{2} : 1 \]