The altitude of a circular cylinder is increased six times and the base area is decreased one-ninth of its value. The factor by which the lateral surface of the cylinder increases, is `2/3` (b) `1/2` (c) `3/2` (d) 2

Explanation

To solve the problem, we need to determine the factor by which the lateral surface area of a circular cylinder increases when the altitude is increased six times and the base area is decreased to one-ninth of its original value.

### Step-by-Step Solution:

1. **Define Variables:**
- Let the original height of the cylinder be \( h_1 \).
- Let the original radius of the base be \( r_1 \).
- The original lateral surface area \( S_1 \) is given by the formula:
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