The total surface area of a cyclinder is ` 6512 cm ^(2)` and the circumference of its base is 88 cm Find
(i) its radius
(ii) its volume.

Answer

To solve the problem step by step, we need to find the radius and volume of the cylinder given its total surface area and the circumference of its base.

### Step 1: Find the radius of the cylinder

We know that the circumference \( C \) of the base of the cylinder is given by the formula:
\[ C = 2\pi R \]
Given that \( C = 88 \, \text{cm} \), we can set up the equation:
\[ 2\pi R = 88 \]

Now, we can solve for \( R \):
\[ R = \frac{88}{2\pi} \]
Using \( \pi \approx \frac{22}{7} \):
\[ R = \frac{88}{2 \times \frac{22}{7}} = \frac{88 \times 7}{44} = \frac{616}{44} = 14 \, \text{cm} \]

### Step 2: Find the height of the cylinder

The total surface area \( A \) of a cylinder is given by the formula:
\[ A = 2\pi R (R + H) \]
Given that \( A = 6512 \, \text{cm}^2 \), we can substitute the known values:
\[ 6512 = 2\pi \times 14 \times (14 + H) \]
Substituting \( \pi \approx \frac{22}{7} \):
\[ 6512 = 2 \times \frac{22}{7} \times 14 \times (14 + H) \]
Calculating \( 2 \times \frac{22}{7} \times 14 \):
\[ = \frac{44 \times 14}{7} = \frac{616}{7} \]
Now substituting back:
\[ 6512 = \frac{616}{7} \times (14 + H) \]
To eliminate the fraction, multiply both sides by 7:
\[ 6512 \times 7 = 616 \times (14 + H) \]
Calculating \( 6512 \times 7 \):
\[ = 45684 \]
Now we have:
\[ 45684 = 616 \times (14 + H) \]
Dividing both sides by 616:
\[ 14 + H = \frac{45684}{616} = 74 \]
Now, solving for \( H \):
\[ H = 74 - 14 = 60 \, \text{cm} \]

### Step 3: Find the volume of the cylinder

The volume \( V \) of a cylinder is given by the formula:
\[ V = \pi R^2 H \]
Substituting the values we found:
\[ V = \pi \times (14)^2 \times 60 \]
Using \( \pi \approx \frac{22}{7} \):
\[ V = \frac{22}{7} \times 196 \times 60 \]
Calculating \( 196 \times 60 \):
\[ = 11760 \]
Now substituting back:
\[ V = \frac{22 \times 11760}{7} \]
Calculating \( \frac{22 \times 11760}{7} \):
\[ = \frac{258720}{7} = 36960 \, \text{cm}^3 \]

### Final Answers:
(i) Radius \( R = 14 \, \text{cm} \)
(ii) Volume \( V = 36960 \, \text{cm}^3 \)

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