I am trying to independently solve physics problems and I am attempting to solve Chapter 5 problem 4-4 on page 140 from the book "Feynman's Tips on Physics".

The problem is as follows:

A painter weighing 180 lbs working from a "bosun's" chair hung down the side of a tall building desires to move in a hurry. He pulls down on a fall rope with such a force that he presses against the chair with only a force of 100 lb. The chair itself weighs 30 lb.

(a) What is the acceleration of the painter and the chair?

(b) What is the total force supported by the pulley?

The diagram is as follows:

MY WORK:

(a) The question states the weights of the painter and the chair are 180 lbs and 30 lbs respectively. I used this idea to perform the following step:

$$\text {Total mass of the painter and the chair} = \left(\frac{(180 + 30) \rm lbs}{g}\right) $$

He exerts a downward force of 100 lb on the chair. His net motion will be upwards.

So the equation I formed is:

$\text{Total force from the upwards motion = Weight of man + Weight of chair + Force on the chair}$

$\frac{180 + 30}{g}\times a = $180 lb + 30 lb + 100 lb

This is giving me:

a = $\frac{31 g}{21} \frac {m}{s^2}$

This answer is incorrect.

I think the 100 lbs force the person exerts on the chair is transferred to the rope he is pulling on.

But that is just the string he is pulling on. The diagram shows that only one string is attached to the bosun chair. That string will have an upwards force of (100 + 180 + 30) lbs (as shown in the picture). This way, one string will have a 100 lbs force and other a force of (100 + 180 + 30) lbs. I don't know if this is even possible.

I do not know how to proceed from this point. How is the 100 lbs downwards force impacting the free body diagram of the entire system?