Let's rephrase the given information and calculations:
We designate the filling rate of pipe A as 'a' and the emptying rate of tank by pipe B as 'b'.
Given the provided data, when pipe A operates from 2 PM to 10 PM (8 hours) and pipe B operates from 3 PM to 10 PM (7 hours), the tank becomes completely filled. Thus, we have:
8a - 7b = 1 [1]
Also, according to the second scenario, when pipe A is open from 2 PM to 6 PM (4 hours) and pipe B operates for 2 hours (from 4 PM to 6 PM), the tank gets filled. This leads to:
4a - 2b = 1 [2]
By multiplying equation [2] and subtracting [1] from it, we obtain:
8a - 4b - (8a - 7b) = 2 - 1
This simplifies to:
7b - 4b = 1
Which means: b = \(\frac{1}{3}\)
Substituting the value of b into equation [2], we get:
\(4a - 2(\frac{1}{3}) = 1\)
This simplifies to:
\(4a -\frac{ 2}{3} = 1\)
So: \(4a = 1 + \frac{2}{3}\)
Which leads to: \(a = \frac{5}{12}\)
Hence, the filling rate is \(\frac{5}{12}.\)
If only pipe A is operational, and the tank gets filled in 'n' hours, we have the equation:
\(n \times a = 1\)
Substituting the value of a:
\(n \times (\frac{5}{12}) = 1\)
This simplifies to: \(n = \frac{12}{5}\)
So, n = 2.4 hours or 144 minutes.