The radius of the base and height of a solid right ciruclar cylinder are in the ratio `2:3` and its volume is 1671 `cm^(3)` .Find the total surface area of the cylinder.

Answer

To find the total surface area of the solid right circular cylinder, we will follow these steps:

### Step 1: Understand the given ratio and assign variables
The radius (r) and height (h) of the cylinder are in the ratio 2:3. We can express them in terms of a variable \( x \):
- Let \( r = 2x \)
- Let \( h = 3x \)

### Step 2: Use the volume formula
The volume \( V \) of a cylinder is given by the formula:
\[ V = \pi r^2 h \]
Substituting the expressions for \( r \) and \( h \):
\[ 1671 = \pi (2x)^2 (3x) \]
This simplifies to:
\[ 1671 = \pi (4x^2)(3x) = 12\pi x^3 \]

### Step 3: Substitute the value of \( \pi \)
Using \( \pi \approx \frac{22}{7} \):
\[ 1671 = 12 \cdot \frac{22}{7} x^3 \]
Multiplying both sides by 7 to eliminate the fraction:
\[ 1671 \cdot 7 = 12 \cdot 22 x^3 \]
Calculating \( 1671 \cdot 7 \):
\[ 11797 = 264 x^3 \]

### Step 4: Solve for \( x^3 \)
Now, divide both sides by 264:
\[ x^3 = \frac{11797}{264} \]
Calculating this gives:
\[ x^3 = 44.67 \quad \text{(approximately)} \]
Taking the cube root:
\[ x \approx \sqrt[3]{44.67} \approx 3.57 \quad \text{(approximately)} \]

### Step 5: Calculate the radius and height
Now we can find \( r \) and \( h \):
- \( r = 2x = 2 \cdot 3.57 \approx 7.14 \, \text{cm} \)
- \( h = 3x = 3 \cdot 3.57 \approx 10.71 \, \text{cm} \)

### Step 6: Calculate the total surface area
The total surface area (TSA) of a cylinder is given by:
\[ TSA = 2\pi r^2 + 2\pi rh \]
Substituting the values of \( r \) and \( h \):
\[ TSA = 2\pi (7.14)^2 + 2\pi (7.14)(10.71) \]
Calculating \( r^2 \):
\[ (7.14)^2 \approx 50.98 \]
Calculating \( rh \):
\[ (7.14)(10.71) \approx 76.34 \]
Now substituting back into the TSA formula:
\[ TSA = 2\pi (50.98) + 2\pi (76.34) \]
\[ TSA = 2\pi (127.32) \approx 2 \cdot \frac{22}{7} \cdot 127.32 \]
Calculating this gives:
\[ TSA \approx 2 \cdot 3.14 \cdot 127.32 \approx 800.56 \, \text{cm}^2 \]

### Final Answer
The total surface area of the cylinder is approximately \( 800.56 \, \text{cm}^2 \).

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