I already have the correct answer to this problem; however, I need it worked aga

Question

I already have the correct answer to this problem; however, I need it worked again step-by-step so I can learn how to do it. Please show work for every step. Thanks! A uniform stationary ladder of length L = 2.6 m and mass M = 19kg leans against a smooth vertical wall, while its bottom legs rest on a rough horizontal floor. The coefficient of static friction between the floor and ladder is mu = 0.43. The ladder makes an angle theta = 57 degrees. with respect to the floor. A painter of mass 8M stands on the ladder a distance d from its base. Find the magnitude of the normal force N, in Newtons, exerted by the floor on the ladder. (N = 1678 is the answer for this part-I need to see step by step how to arrive at this) What is the largest distance up the ladder d_max, in meters, that the painter can stand without the ladder slipping? (d_max = 1.774 is the correct answer; I need to see step-by-step how to arrive at this answer)

Explanation / Answer

We have to consider

Sum of the vertical force down =Normal force on the bottom of a ladder

Weight of ladder =19kg*9.81m/s2 =186.4N

Weight of the painter =19kg*8*9.81 =1491.12N

Now the normal force is =186.4+1491.12 =1677.52N

B)

At the threshold FN =NW

Therefore NW = 1/2Mg cottheta (1+16d/L)

FN =0.43*1677.52N =721.33

Now substitute the values in the above equation to find d

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