50 circular paltes each of radius 7 cm and thickness 0.5 cm are placed one above the other to form a solid right circular cylinder .Find (i) the total surface area and (ii) the volume of the cylinder so formed.

Explanation

To solve the problem, we will follow these steps:

### Given Data:
- Number of plates (n) = 50
- Radius of each plate (r) = 7 cm
- Thickness of each plate (t) = 0.5 cm

### Step 1: Calculate the height of the cylinder
The height of the cylinder (h) is the total thickness of all the plates stacked together.

\[ h = n \times t = 50 \times 0.5 = 25 \text{ cm} \]

### Step 2: Calculate the total surface area of the cylinder
The formula for the total surface area (TSA) of a cylinder is given by:

\[ \text{TSA} = 2\pi r (r + h) \]

Substituting the values of \( r \) and \( h \):

\[ \text{TSA} = 2 \times \frac{22}{7} \times 7 \times (7 + 25) \]

Calculating \( 7 + 25 \):

\[ 7 + 25 = 32 \]

Now substituting back into the TSA formula:

\[ \text{TSA} = 2 \times \frac{22}{7} \times 7 \times 32 \]

The \( 7 \) in the numerator and denominator cancels out:

\[ \text{TSA} = 2 \times 22 \times 32 \]

Calculating \( 2 \times 22 = 44 \):

\[ \text{TSA} = 44 \times 32 = 1408 \text{ cm}^2 \]

### Step 3: Calculate the volume of the cylinder
The formula for the volume (V) of a cylinder is given by:

\[ V = \pi r^2 h \]

Substituting the values of \( r \) and \( h \):

\[ V = \frac{22}{7} \times 7^2 \times 25 \]

Calculating \( 7^2 \):

\[ 7^2 = 49 \]

Now substituting back into the volume formula:

\[ V = \frac{22}{7} \times 49 \times 25 \]

The \( 7 \) in the denominator cancels with \( 49 \):

\[ V = 22 \times 7 \times 25 \]

Calculating \( 22 \times 7 = 154 \):

\[ V = 154 \times 25 \]

Calculating \( 154 \times 25 \):

\[ V = 3850 \text{ cm}^3 \]

### Final Answers:
(i) Total Surface Area = 1408 cm²
(ii) Volume = 3850 cm³