A solid right circular cylinder and a solid hemisphere stand on equal bases and have the same height. The ratio of their whole surface area is :

Options

A. `3:2`
B. `3:4`
C. `4:3`
D. `2:3`

Answer

To find the ratio of the whole surface area of a solid right circular cylinder and a solid hemisphere that stand on equal bases and have the same height, we can follow these steps:

### Step 1: Define the dimensions
Let the radius of both the cylinder and the hemisphere be \( r \). Since they have the same height, the height of the cylinder \( h \) is also equal to \( r \).

### Step 2: Calculate the surface area of the cylinder
The formula for the whole surface area \( A_c \) of a right circular cylinder is given by:
\[ A_c = 2\pi r h + 2\pi r^2 \]
Here, \( h = r \), so we substitute \( h \) into the formula:
\[ A_c = 2\pi r (r) + 2\pi r^2 = 2\pi r^2 + 2\pi r^2 = 4\pi r^2 \]

### Step 3: Calculate the surface area of the hemisphere
The formula for the whole surface area \( A_h \) of a solid hemisphere is given by:
\[ A_h = 3\pi r^2 \]

### Step 4: Set up the ratio of the surface areas
Now, we can set up the ratio of the surface areas of the cylinder to the hemisphere:
\[ \text{Ratio} = \frac{A_c}{A_h} = \frac{4\pi r^2}{3\pi r^2} \]

### Step 5: Simplify the ratio
The \( \pi r^2 \) terms cancel out:
\[ \text{Ratio} = \frac{4}{3} \]

### Final Answer
Thus, the ratio of the whole surface area of the solid right circular cylinder to that of the solid hemisphere is:
\[ \frac{4}{3} \]

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