The ratio of diameter and height of a right circular cylinder is 4: 3. If diameter of the cylinder get reduced by 25% then its total surface area reduced to `416 pi` square meter. What is the circumference of the base of the cylinder.

Options

A. `32 pi cm`
B. `44 pi cm`
C. `35 pi cm`
D. `7 pi cm`

Explanation

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

### Step 1: Define the variables based on the ratio
The ratio of the diameter to the height of the cylinder is given as 4:3.
Let the diameter be \( 4x \) and the height be \( 3x \).

### Step 2: Calculate the radius
The radius \( r \) of the cylinder can be calculated as:
\[ r = \frac{\text{Diameter}}{2} = \frac{4x}{2} = 2x \]

### Step 3: Calculate the new diameter after reduction
The diameter is reduced by 25%. Therefore, the new diameter \( D' \) can be calculated as:
\[ D' = 4x - 0.25 \times 4x = 4x - x = 3x \]

### Step 4: Calculate the new radius
The new radius \( r' \) is:
\[ r' = \frac{D'}{2} = \frac{3x}{2} = 1.5x \]

### Step 5: Calculate the total surface area before and after the reduction
The total surface area \( A \) of a cylinder is given by the formula:
\[ A = 2\pi r (r + h) \]
Substituting the original radius and height:
\[ A = 2\pi (2x) \left(2x + 3x\right) = 2\pi (2x)(5x) = 20\pi x^2 \]

The total surface area after the diameter reduction becomes:
\[ A' = 2\pi r' (r' + h) = 2\pi (1.5x) \left(1.5x + 3x\right) = 2\pi (1.5x)(4.5x) = 13.5\pi x^2 \]

### Step 6: Set up the equation for the reduction in total surface area
According to the problem, the total surface area is reduced to \( 416\pi \):
\[ 20\pi x^2 - 13.5\pi x^2 = 416\pi \]
This simplifies to:
\[ 6.5\pi x^2 = 416\pi \]

### Step 7: Solve for \( x^2 \)
Dividing both sides by \( \pi \):
\[ 6.5x^2 = 416 \]
Now, divide by 6.5:
\[ x^2 = \frac{416}{6.5} = 64 \]

### Step 8: Calculate \( x \)
Taking the square root of both sides:
\[ x = 8 \]

### Step 9: Calculate the radius
Now, substituting back to find the radius:
\[ r = 2x = 2 \times 8 = 16 \text{ meters} \]

### Step 10: Calculate the circumference of the base
The circumference \( C \) of the base of the cylinder is given by:
\[ C = 2\pi r = 2\pi (16) = 32\pi \text{ meters} \]

### Final Answer
The circumference of the base of the cylinder is \( 32\pi \) meters.

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