A sphere is melted and molded into solid cylinder. If radius of both solids is equal, then find the ratio of total surface area of sphere to the total surface area of cylinder

Options

A. `2:3`
B. `4:3`
C. `3:7`
D. `6:7`

Answer

To find the ratio of the total surface area of a sphere to the total surface area of a cylinder, we will follow these steps:

### Step 1: Write the formula for the volume of a sphere
The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]

### Step 2: Write the formula for the volume of a cylinder
The volume \( V \) of a cylinder is given by the formula:
\[ V = \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height of the cylinder.

### Step 3: Set the volumes equal to each other
Since the sphere is melted and molded into a cylinder, their volumes are equal:
\[ \frac{4}{3} \pi r^3 = \pi r^2 h \]

### Step 4: Simplify the equation
We can cancel \( \pi \) from both sides:
\[ \frac{4}{3} r^3 = r^2 h \]
Now, divide both sides by \( r^2 \) (assuming \( r \neq 0 \)):
\[ \frac{4}{3} r = h \]

### Step 5: Write the formula for the total surface area of a sphere
The total surface area \( A_s \) of a sphere is given by:
\[ A_s = 4 \pi r^2 \]

### Step 6: Write the formula for the total surface area of a cylinder
The total surface area \( A_c \) of a cylinder is given by:
\[ A_c = 2 \pi r (r + h) \]

### Step 7: Substitute \( h \) in the cylinder's surface area formula
From Step 4, we found \( h = \frac{4}{3} r \). Substitute this value into the cylinder's surface area formula:
\[ A_c = 2 \pi r \left(r + \frac{4}{3} r\right) \]
\[ A_c = 2 \pi r \left(\frac{3r + 4r}{3}\right) = 2 \pi r \left(\frac{7r}{3}\right) = \frac{14 \pi r^2}{3} \]

### Step 8: Find the ratio of the total surface areas
Now we can find the ratio of the total surface area of the sphere to that of the cylinder:
\[ \text{Ratio} = \frac{A_s}{A_c} = \frac{4 \pi r^2}{\frac{14 \pi r^2}{3}} \]

### Step 9: Simplify the ratio
Cancel \( \pi r^2 \) from the numerator and denominator:
\[ \text{Ratio} = \frac{4}{\frac{14}{3}} = 4 \cdot \frac{3}{14} = \frac{12}{14} = \frac{6}{7} \]

### Final Answer
The ratio of the total surface area of the sphere to the total surface area of the cylinder is:
\[ \boxed{6:7} \]