7.
\includegraphics[max width=\textwidth, alt={}]{baa3525f-6263-4dec-b581-531d8258a2e1-2_332_410_1837_402} A uniform ladder \(A B\), of mass \(m \mathrm {~kg}\) and length \(2 a \mathrm {~m}\), rests with its upper end \(A\) in contact with a smooth vertical wall and its lower end \(B\) in contact with a fixed peg on horizontal ground. The ladder makes an angle \(\alpha\) with the ground, where \(\tan \alpha = \frac { 3 } { 4 }\).

1. Show that the magnitude of the resultant force acting on the ladder at \(B\) is \(\frac { \sqrt { } 13 } { 3 } \mathrm { mg }\).
2. Find, to the nearest degree, the direction of this resultant force at \(B\). The peg will break when the horizontal force acting on it exceeds \(2 m g \mathrm {~N}\). A painter of mass \(6 m \mathrm {~kg}\) starts to climb the ladder from \(B\).
3. Find, in terms of \(a\), the greatest distance up the ladder that the painter can safely climb.