A solid hemisphere has radius 14 cm. It is melted to form a cylinder such that the ratio of its curved surface area and total surface area is 2 : 3. What is the radius (in cm) of its base?

Options

A. `(10)/(root(3)(3))`
B. `(14)/(root(3)(3))`
C. `(7)/(root(3)(3))`
D. `(21)/(root(3)(3))`

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the given information
We have a solid hemisphere with a radius of 14 cm. When melted, it forms a cylinder. The ratio of the curved surface area (CSA) to the total surface area (TSA) of the cylinder is given as 2:3.

### Step 2: Calculate the volume of the hemisphere
The volume \( V \) of a hemisphere is given by the formula:
\[ V = \frac{2}{3} \pi r^3 \]
Substituting \( r = 14 \) cm:
\[ V = \frac{2}{3} \pi (14)^3 \]

### Step 3: Calculate the volume of the cylinder
The volume \( V \) of a cylinder is given by the formula:
\[ V = \pi r^2 h \]
where \( r \) is the radius of the base and \( h \) is the height of the cylinder.

### Step 4: Set the volumes equal
Since the hemisphere is melted to form the cylinder, we can set the volumes equal:
\[ \frac{2}{3} \pi (14)^3 = \pi r^2 h \]
We can cancel \( \pi \) from both sides:
\[ \frac{2}{3} (14)^3 = r^2 h \]

### Step 5: Express the ratio of CSA and TSA
The curved surface area (CSA) of the cylinder is given by:
\[ CSA = 2 \pi r h \]
The total surface area (TSA) of the cylinder is given by:
\[ TSA = 2 \pi r h + 2 \pi r^2 \]
The ratio of TSA to CSA is given as:
\[ \frac{TSA}{CSA} = \frac{2 \pi r h + 2 \pi r^2}{2 \pi r h} = \frac{h + r}{h} \]
Given the ratio is \( \frac{3}{2} \):
\[ \frac{h + r}{h} = \frac{3}{2} \]

### Step 6: Solve for \( h \) in terms of \( r \)
Cross-multiplying gives:
\[ 2(h + r) = 3h \]
This simplifies to:
\[ 2h + 2r = 3h \implies h = 2r \]

### Step 7: Substitute \( h \) back into the volume equation
Substituting \( h = 2r \) into the volume equation:
\[ \frac{2}{3} (14)^3 = r^2 (2r) \]
This simplifies to:
\[ \frac{2}{3} (14)^3 = 2r^3 \]
Dividing both sides by 2:
\[ \frac{1}{3} (14)^3 = r^3 \]

### Step 8: Solve for \( r \)
Multiplying both sides by 3:
\[ (14)^3 = 3r^3 \]
Taking the cube root:
\[ r = \frac{14}{\sqrt[3]{3}} \]

### Final Answer
Thus, the radius of the base of the cylinder is:
\[ r = \frac{14}{\sqrt[3]{3}} \text{ cm} \]