The total surface area of a right circular cylinder (A) is 200 `pi``cm^2` and the respectively ratio between its diameter and height is 2:3. If a square’s side is equal to A’s height, what will be the square’s perimeter? (in cm)

Options

A. 60
B. 80
C. 20
D. None of these

Answer

To solve the problem step by step, we will follow the given information and apply the relevant formulas.

### Step 1: Understand the given information
We know the total surface area \( A \) of a right circular cylinder is \( 200 \pi \, \text{cm}^2 \) and the ratio of its diameter to height is \( 2:3 \).

### Step 2: Write the formula for the total surface area of a cylinder
The total surface area \( A \) of a right circular cylinder is given by the formula:
\[ A = 2\pi r(h + r) \]
where \( r \) is the radius and \( h \) is the height of the cylinder.

### Step 3: Express diameter and height in terms of a variable
Let the common ratio factor be \( k \). Then:
- Diameter \( d = 2r = 2k \)
- Height \( h = 3k \)

From the diameter, we can find the radius:
\[ r = \frac{d}{2} = k \]

### Step 4: Substitute \( r \) and \( h \) into the surface area formula
Substituting \( r \) and \( h \) into the surface area formula:
\[ 200\pi = 2\pi k(3k + k) \]
This simplifies to:
\[ 200\pi = 2\pi k(4k) \]

### Step 5: Simplify the equation
Dividing both sides by \( 2\pi \):
\[ 100 = 4k^2 \]
Now, divide both sides by 4:
\[ 25 = k^2 \]

### Step 6: Solve for \( k \)
Taking the square root of both sides:
\[ k = 5 \]

### Step 7: Find the height of the cylinder
Now, we can find the height \( h \):
\[ h = 3k = 3 \times 5 = 15 \, \text{cm} \]

### Step 8: Find the side of the square
The side of the square is equal to the height of the cylinder, so:
\[ \text{Side of the square} = h = 15 \, \text{cm} \]

### Step 9: Calculate the perimeter of the square
The perimeter \( P \) of a square is given by:
\[ P = 4 \times \text{side} \]
Substituting the side length:
\[ P = 4 \times 15 = 60 \, \text{cm} \]

### Final Answer
The perimeter of the square is \( 60 \, \text{cm} \).

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