It costs Rs 3300 to paint the inner curved surface of a cylindrical vessel 10m deep at the rate of Rs 30 per `m^(2)`. Find the
(i) inner curved surface area of the vessel,
(ii) inner radius of the base, and
(iii) capacity of the vessel

Explanation

To solve the problem step by step, we will follow the instructions given in the video transcript and derive the necessary values.

### Given:
- Cost to paint the inner curved surface = Rs 3300
- Rate of painting = Rs 30 per m²
- Height of the cylindrical vessel (h) = 10 m

### (i) Find the inner curved surface area of the vessel.

1. **Calculate the inner curved surface area (CSA)**:
- The formula for the curved surface area of a cylinder is given by:
\[ \text{CSA} = 2 \pi r h \]
- First, we need to find the total area that was painted. We can calculate it using the cost and rate:
\[ \text{Total area} = \frac{\text{Total cost}}{\text{Rate per m}^2} = \frac{3300}{30} = 110 \, \text{m}^2 \]
- Therefore, the inner curved surface area of the vessel is:
\[ \text{CSA} = 110 \, \text{m}^2 \]

### (ii) Find the inner radius of the base.

2. **Use the CSA to find the radius**:
- We know the CSA and the height, so we can rearrange the CSA formula to find the radius:
\[ 110 = 2 \pi r h \]
- Substitute \( h = 10 \, \text{m} \) and \( \pi \approx \frac{22}{7} \):
\[ 110 = 2 \times \frac{22}{7} \times r \times 10 \]
- Simplifying this gives:
\[ 110 = \frac{440}{7} r \]
- To isolate \( r \), we multiply both sides by \( \frac{7}{440} \):
\[ r = \frac{110 \times 7}{440} = \frac{770}{440} = \frac{77}{44} = 1.75 \, \text{m} \]

### (iii) Find the capacity of the vessel.

3. **Calculate the volume (capacity) of the vessel**:
- The volume \( V \) of a cylinder is given by:
\[ V = \pi r^2 h \]
- Substitute \( r = 1.75 \, \text{m} \) and \( h = 10 \, \text{m} \):
\[ V = \frac{22}{7} \times (1.75)^2 \times 10 \]
- Calculate \( (1.75)^2 = 3.0625 \):
\[ V = \frac{22}{7} \times 3.0625 \times 10 \]
- Now calculate:
\[ V = \frac{22 \times 3.0625 \times 10}{7} = \frac{673.75}{7} \approx 96.25 \, \text{m}^3 \]

### Summary of Results:
- (i) Inner curved surface area = 110 m²
- (ii) Inner radius of the base = 1.75 m
- (iii) Capacity of the vessel = 96.25 m³