I was visiting a family member in the hospital. They had their head elevated in the bed but had slid to the foot of the bed. Because of this, the nurses would come in and slide them back to the head of the bed. Each group nurses did it in a different way. The first set of nurses laid the bed perfectly flat and pulled the patient $($actually the sheet the patient was on$)$ parallel to the mattress. The second set of nurses came in$,$ but first tilted the head of the bed down so it was $10$ degrees below the horizontal and then like the first group pulled parallel to the mattress. To make a problem out of this, lets assume the patient has a mass of $100$ kg and the coefficient of kinetic friction is $0\mathrm{.}8.$

a$)$With what force do the nurses need to pull the patient while moving at constant speed when the bed is flat$/$ horizonal. Draw a force diagram, write Newtons second law applied to the forces in this problem and then solve the problem.

b$)$With what force do the nurses need to pull the patient while moving at a constant speed when the head the bed is tilted down at a $10$ degree angle. Draw a force diagram, write Newtons second law applied to the forces in this problem and then solve the problem.

c$)$When the head of the board is tilted down, by what factor do the nurses in part b$.)$ pull compared to part a$.).$ Give your answer as a percentage i$.$e$.,75\%.$

d$)$Give all the reasons why the pulling force in part b is less than in part a$.$ There are at least two reasons.