The total surface area of a hollow cylinder, which is open from both the sides, is 3575 cm", area of its base ring is `357.5 cm^(2)` and its height is 14 cm. Find the thickness of the cylinder.

Answer

To solve the problem of finding the thickness of a hollow cylinder, we will follow these steps:

### Step 1: Understand the Given Information
We are given:
- Total Surface Area (TSA) of the hollow cylinder = 3575 cm²
- Area of the base ring = 357.5 cm²
- Height (h) of the cylinder = 14 cm

### Step 2: Set Up the Equations
1. The area of the base ring can be expressed as:
\[ \text{Area of base ring} = \pi (R^2 - r^2) = 357.5 \text{ cm}^2 \]
where \( R \) is the outer radius and \( r \) is the inner radius.

2. The total surface area of the hollow cylinder (open from both sides) is given by:
\[ \text{TSA} = 2\pi Rh + 2\pi rh + \text{Area of base ring} \]
Since the cylinder is open from both sides, we can simplify it to:
\[ \text{TSA} = 2\pi h(R + r) + \pi (R^2 - r^2) \]
Setting this equal to the given TSA:
\[ 3575 = 2\pi h(R + r) + 357.5 \]

### Step 3: Substitute Known Values
Substituting \( h = 14 \) cm and rearranging the equation:
\[ 3575 - 357.5 = 2\pi(14)(R + r) \]
\[ 3217.5 = 28\pi(R + r) \]

### Step 4: Calculate \( R + r \)
Now, we can solve for \( R + r \):
\[ R + r = \frac{3217.5}{28\pi} \]
Using \( \pi \approx \frac{22}{7} \):
\[ R + r = \frac{3217.5 \times 7}{28 \times 22} = \frac{22522.5}{616} \approx 36.5 \text{ cm} \]

### Step 5: Solve for \( R^2 - r^2 \)
From the area of the base ring:
\[ R^2 - r^2 = \frac{357.5 \times 7}{22} = \frac{2502.5}{22} \approx 113.75 \text{ cm}^2 \]

### Step 6: Use the Two Equations
Now we have two equations:
1. \( R + r = 36.5 \)
2. \( R^2 - r^2 = 113.75 \)

We can express \( R \) in terms of \( r \):
\[ R = 36.5 - r \]
Substituting into the second equation:
\[ (36.5 - r)^2 - r^2 = 113.75 \]
Expanding and simplifying:
\[ 1332.25 - 73r + r^2 - r^2 = 113.75 \]
\[ 1332.25 - 73r = 113.75 \]
\[ 73r = 1218.5 \]
\[ r = \frac{1218.5}{73} \approx 16.7 \text{ cm} \]

### Step 7: Find \( R \)
Now substituting back to find \( R \):
\[ R = 36.5 - 16.7 \approx 19.8 \text{ cm} \]

### Step 8: Find the Thickness
The thickness \( t \) of the cylinder is given by:
\[ t = R - r = 19.8 - 16.7 \approx 3.1 \text{ cm} \]

### Final Answer
The thickness of the hollow cylinder is approximately **3.1 cm**.

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