Question

Why did they choose x for unknown?

Answer 1

Let's realize that by applying a force to the rope to accelerate the person upwards from the bottom of the cliff, we necessarily have an increase in tension. That is the main idea here. We need to find this maximum acceleration such that a tension of 865N is realized. Once the acceleration is determined, we can use our basic kinematic relations to determine the time it will take to raise the person. To get your started,
\[T = mg + ma\]where \(m\) is the mass of the person and \(a\) is the acceleration we create on the person in raising them upwards.