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. `7:6`
C. `3:7`
D. `6:7`

Answer

To solve the problem of finding the ratio of the total surface area of a sphere to that of a cylinder when both have the same radius, we can follow these steps:

### Step 1: Understand the volumes of the sphere and cylinder
The volume of a sphere is given by the formula:
\[ V_{sphere} = \frac{4}{3} \pi r^3 \]

The volume of a cylinder is given by the formula:
\[ V_{cylinder} = \pi r^2 h \]

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

### Step 2: Simplify the volume equation
We can cancel \(\pi\) from both sides:
\[ \frac{4}{3} r^3 = r^2 h \]

Next, divide both sides by \(r^2\) (assuming \(r \neq 0\)):
\[ \frac{4}{3} r = h \]

### Step 3: Write the total surface area formulas
The total surface area of the sphere is given by:
\[ A_{sphere} = 4 \pi r^2 \]

The total surface area of the cylinder is given by:
\[ A_{cylinder} = 2 \pi r (r + h) \]

### Step 4: Substitute \(h\) into the cylinder's surface area formula
Substituting \(h = \frac{4}{3} r\) into the cylinder's surface area formula:
\[ A_{cylinder} = 2 \pi r \left(r + \frac{4}{3} r\right) \]
\[ = 2 \pi r \left(\frac{3r + 4r}{3}\right) \]
\[ = 2 \pi r \left(\frac{7r}{3}\right) \]
\[ = \frac{14 \pi r^2}{3} \]

### Step 5: Find the ratio of the surface areas
Now, we can find the ratio of the total surface area of the sphere to that of the cylinder:
\[ \text{Ratio} = \frac{A_{sphere}}{A_{cylinder}} = \frac{4 \pi r^2}{\frac{14 \pi r^2}{3}} \]

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

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