The ratio of the diameter to the height of a right circular cylinder is 4:3 If the diameter of the cylinder gets reduced by 25% then its total surface area is reduced to 2079 square metre. What is the circumference of the base of the original cylinder? (in m)

Options

A. `14pi`
B. `7sqrtpi`
C. `7pi`
D. `28pi`

Answer

To solve the problem, we need to follow these steps:

### Step 1: Define the Variables
Let the diameter of the original cylinder be \( D \) and the height be \( H \). According to the problem, the ratio of the diameter to the height is given as \( 4:3 \).

We can express the diameter and height in terms of a variable \( k \):
- Diameter \( D = 4k \)
- Height \( H = 3k \)

### Step 2: Calculate the Radius
The radius \( r \) of the cylinder is half of the diameter:
\[ r = \frac{D}{2} = \frac{4k}{2} = 2k \]

### Step 3: Calculate the Reduced Diameter
The diameter is reduced by 25%, so the new diameter \( D' \) is:
\[ D' = D - 0.25D = D(1 - 0.25) = D \times 0.75 = 4k \times 0.75 = 3k \]

The new radius \( r' \) becomes:
\[ r' = \frac{D'}{2} = \frac{3k}{2} = 1.5k \]

### Step 4: Calculate the New Height
The height remains the same as the original height:
\[ H' = H = 3k \]

### Step 5: Calculate the Total Surface Area
The total surface area \( A \) of a cylinder is given by the formula:
\[ A = 2\pi r^2 + 2\pi r H \]
For the original cylinder, the total surface area \( A \) is:
\[ A = 2\pi (2k)^2 + 2\pi (2k)(3k) = 2\pi (4k^2) + 2\pi (6k^2) = 8\pi k^2 + 12\pi k^2 = 20\pi k^2 \]

For the reduced cylinder, the total surface area \( A' \) is:
\[ A' = 2\pi (1.5k)^2 + 2\pi (1.5k)(3k) = 2\pi (2.25k^2) + 2\pi (4.5k^2) = 4.5\pi k^2 + 9\pi k^2 = 13.5\pi k^2 \]

### Step 6: Set Up the Equation
According to the problem, the total surface area of the reduced cylinder is given as 2079 square meters:
\[ 13.5\pi k^2 = 2079 \]

### Step 7: Solve for \( k^2 \)
To find \( k^2 \), we can rearrange the equation:
\[ k^2 = \frac{2079}{13.5\pi} \]

### Step 8: Calculate \( k \)
Now, we can calculate \( k \):
\[ k^2 = \frac{2079}{13.5 \times 3.14} \approx \frac{2079}{42.39} \approx 49 \]
\[ k \approx 7 \]

### Step 9: Find the Circumference of the Original Cylinder
The circumference \( C \) of the base of the original cylinder is given by:
\[ C = \pi D = \pi (4k) = 4\pi k \]
Substituting \( k \):
\[ C = 4\pi \times 7 = 28\pi \approx 87.96 \text{ meters} \]

### Final Answer
The circumference of the base of the original cylinder is approximately \( 87.96 \) meters.