Radius of hemisphere is twice that of a sphere. What is the ratio of total surface area of hemisphere and sphere ?

Options

A. `3:1`
B. `12:1`
C. `4:1`
D. `6:1`

Answer

To find the ratio of the total surface area of a hemisphere to that of a sphere when the radius of the hemisphere is twice that of the sphere, we can follow these steps:

### Step 1: Define the Radii
Let the radius of the sphere be \( r \). Therefore, the radius of the hemisphere will be \( 2r \) (since it is twice that of the sphere).

### Step 2: Calculate the Total Surface Area of the Sphere
The formula for the total surface area of a sphere is given by:
\[ \text{Total Surface Area of Sphere} = 4\pi r^2 \]

### Step 3: Calculate the Total Surface Area of the Hemisphere
The formula for the total surface area of a hemisphere is:
\[ \text{Total Surface Area of Hemisphere} = 3\pi R^2 \]
where \( R \) is the radius of the hemisphere. Since \( R = 2r \), we substitute this into the formula:
\[ \text{Total Surface Area of Hemisphere} = 3\pi (2r)^2 = 3\pi (4r^2) = 12\pi r^2 \]

### Step 4: Find the Ratio of the Total Surface Areas
Now we can find the ratio of the total surface area of the hemisphere to that of the sphere:
\[ \text{Ratio} = \frac{\text{Total Surface Area of Hemisphere}}{\text{Total Surface Area of Sphere}} = \frac{12\pi r^2}{4\pi r^2} \]

### Step 5: Simplify the Ratio
We can simplify this ratio:
\[ \text{Ratio} = \frac{12\pi r^2}{4\pi r^2} = \frac{12}{4} = 3 \]

### Final Answer
Thus, the ratio of the total surface area of the hemisphere to that of the sphere is:
\[ \text{Ratio} = 3:1 \]

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