A cone , a hemisphere and a cylinder stand on equal bases of radius R and have equal heights H . Their whole surfaces are in the ratio :

Options

A. a)`(sqrt(3) +1): 3:4`
B. b)`(sqrt(2) + 1) : 7 : 8 `
C. c)`(sqrt(2) + 1) : 3: 4`
D. d)none of these

Answer

To find the ratio of the whole surface areas of a cone, a hemisphere, and a cylinder that stand on equal bases of radius \( R \) and have equal heights \( H \), we will calculate the surface areas of each shape step by step.

### Step 1: Calculate the Surface Area of the Cone

The formula for the total surface area of a cone is given by:

\[ \text{Surface Area of Cone} = \pi R l + \pi R^2 \]

where \( l \) is the slant height of the cone. The slant height can be calculated using the Pythagorean theorem:

\[ l = \sqrt{H^2 + R^2} \]

Thus, the surface area of the cone becomes:

\[ \text{Surface Area of Cone} = \pi R \sqrt{H^2 + R^2} + \pi R^2 \]

### Step 2: Calculate the Surface Area of the Hemisphere

The formula for the total surface area of a hemisphere is:

\[ \text{Surface Area of Hemisphere} = 2\pi R^2 + \pi R^2 = 3\pi R^2 \]

### Step 3: Calculate the Surface Area of the Cylinder

The formula for the total surface area of a cylinder is:

\[ \text{Surface Area of Cylinder} = 2\pi R H + 2\pi R^2 \]

### Step 4: Substitute \( R \) and \( H \)

Since the problem states that the cone, hemisphere, and cylinder have equal bases of radius \( R \) and equal heights \( H \), we will substitute these values into our equations.

1. For the cone:

\[ \text{Surface Area of Cone} = \pi R \sqrt{H^2 + R^2} + \pi R^2 \]

2. For the hemisphere:

\[ \text{Surface Area of Hemisphere} = 3\pi R^2 \]

3. For the cylinder:

\[ \text{Surface Area of Cylinder} = 2\pi R H + 2\pi R^2 \]

### Step 5: Find the Ratio of the Surface Areas

Now we will express the surface areas in terms of \( R \) and \( H \) and find the ratio:

Let’s denote:
- \( A_c \) = Surface Area of Cone
- \( A_h \) = Surface Area of Hemisphere
- \( A_{cy} \) = Surface Area of Cylinder

Thus, we have:

\[ A_c = \pi R \sqrt{H^2 + R^2} + \pi R^2 \]
\[ A_h = 3\pi R^2 \]
\[ A_{cy} = 2\pi R H + 2\pi R^2 \]

### Step 6: Simplify the Ratios

To find the ratio, we can factor out \( \pi \):

\[ \text{Ratio} = A_c : A_h : A_{cy} = \left( R \sqrt{H^2 + R^2} + R^2 \right) : 3R^2 : (2RH + 2R^2) \]

Now, we can express everything in terms of \( R \) and \( H \) and simplify:

1. For the cone, we can substitute \( H = R \) (as given):
\[ A_c = \pi R \sqrt{R^2 + R^2} + \pi R^2 = \pi R \sqrt{2R^2} + \pi R^2 = \pi R \cdot R\sqrt{2} + \pi R^2 = \pi R^2(\sqrt{2} + 1) \]

2. For the hemisphere:
\[ A_h = 3\pi R^2 \]

3. For the cylinder:
\[ A_{cy} = 2\pi R^2 + 2\pi R^2 = 4\pi R^2 \]

### Final Ratio

Thus, the ratio of the surface areas becomes:

\[ \text{Ratio} = (\sqrt{2} + 1) : 3 : 4 \]