The curved surface area of a cyclinder is ` 17,600 cm ^(2) ` and the circumference of its base is 220 cm Find :
(i) the height of the cyclinder
(ii) the volume of the cyclinder.

Answer

To solve the problem step by step, we will find the height of the cylinder first and then calculate its volume.

### Step 1: Find the radius of the cylinder
We know the circumference of the base of the cylinder is given by the formula:
\[ C = 2\pi r \]
Given that the circumference \(C = 220 \, \text{cm}\), we can set up the equation:
\[ 2\pi r = 220 \]
Substituting \(\pi\) with \(\frac{22}{7}\):
\[ 2 \times \frac{22}{7} \times r = 220 \]
Now, simplifying this:
\[ \frac{44}{7} r = 220 \]
To isolate \(r\), multiply both sides by \(\frac{7}{44}\):
\[ r = 220 \times \frac{7}{44} \]
Calculating this gives:
\[ r = 220 \times \frac{7}{44} = 35 \, \text{cm} \]

### Step 2: Find the height of the cylinder
The curved surface area (CSA) of the cylinder is given by the formula:
\[ \text{CSA} = 2\pi rh \]
We know the CSA is \(17600 \, \text{cm}^2\). Substituting the values we have:
\[ 17600 = 2 \times \frac{22}{7} \times 35 \times h \]
Calculating \(2 \times \frac{22}{7} \times 35\):
\[ 2 \times \frac{22}{7} \times 35 = \frac{1540}{7} \]
Now substituting back into the equation:
\[ 17600 = \frac{1540}{7} \times h \]
To solve for \(h\), multiply both sides by \(\frac{7}{1540}\):
\[ h = 17600 \times \frac{7}{1540} \]
Calculating this gives:
\[ h = 17600 \times \frac{7}{1540} = 80 \, \text{cm} \]

### Step 3: Find the volume of the cylinder
The volume \(V\) of the cylinder is given by the formula:
\[ V = \pi r^2 h \]
Substituting the known values:
\[ V = \frac{22}{7} \times 35^2 \times 80 \]
Calculating \(35^2\):
\[ 35^2 = 1225 \]
Now substituting this back into the volume formula:
\[ V = \frac{22}{7} \times 1225 \times 80 \]
Calculating \(1225 \times 80\):
\[ 1225 \times 80 = 98000 \]
Now substituting this back into the volume formula:
\[ V = \frac{22}{7} \times 98000 \]
Calculating this gives:
\[ V = 22 \times 14000 = 308000 \, \text{cm}^3 \]

### Final Answers:
(i) The height of the cylinder is \(80 \, \text{cm}\).

(ii) The volume of the cylinder is \(308000 \, \text{cm}^3\).