The ratio of the total surface area to the lateral surface area of the cylinder is `5:3` Find the height of the cylinder if the radius of the cylinder is 12 cm ?

Answer

To solve the problem, we need to find the height of a cylinder given the ratio of its total surface area to its lateral surface area and the radius. Here’s a step-by-step solution:

### Step 1: Understand the formulas
The total surface area (TSA) of a cylinder is given by:
\[ \text{TSA} = 2\pi r(h + r) \]
The lateral surface area (LSA) of a cylinder is given by:
\[ \text{LSA} = 2\pi rh \]

### Step 2: Set up the ratio
According to the problem, the ratio of the total surface area to the lateral surface area is given as:
\[ \frac{\text{TSA}}{\text{LSA}} = \frac{5}{3} \]

### Step 3: Substitute the formulas into the ratio
Substituting the formulas for TSA and LSA into the ratio gives us:
\[ \frac{2\pi r(h + r)}{2\pi rh} = \frac{5}{3} \]
The \(2\pi\) cancels out:
\[ \frac{r(h + r)}{rh} = \frac{5}{3} \]

### Step 4: Simplify the equation
We can simplify this equation:
\[ \frac{h + r}{h} = \frac{5}{3} \]
Cross-multiplying gives:
\[ 3(h + r) = 5h \]

### Step 5: Rearrange the equation
Expanding and rearranging the equation:
\[ 3h + 3r = 5h \]
\[ 3r = 5h - 3h \]
\[ 3r = 2h \]

### Step 6: Solve for height (h)
Now, we can express height in terms of radius:
\[ h = \frac{3r}{2} \]

### Step 7: Substitute the radius value
We know the radius \(r = 12 \, \text{cm}\):
\[ h = \frac{3 \times 12}{2} = \frac{36}{2} = 18 \, \text{cm} \]

### Final Answer
The height of the cylinder is:
\[ \boxed{18 \, \text{cm}} \]