A construction Worker

Question

A construction worker pulls a five-meter plank up the side of a building under construction by means of a rope tied to one end of the plank (see figure). Assume the opposite end of the plank follows a path perpendicular to the wall of the building and the worker pulls the rope at a rate of 0.22 meters per second. How fast is the end of the plank sliding along the ground when it is 2.3 meters from the wall of the building? (Round your answer to two decimal places.)

Answer 1

let a = the height of the plank, b = distance of the end of the plank on the ground to the wall = 2.3 m. a' = .22 m/sec a=sqr(25-2.3^2) = sqr(25-5.29) = sqr(19.71) = 4.43

plug in a=4.43, b=2.3, a'=.22, to solve for b'

a^2 + b^2 = 5^2

2aa' + 2bb' =0

aa' + bb' = 0

b' = -aa'/b = -4.43(.22)/2.3 = -0.43 m/sec at an instant in time when the plank is 2.3 meters from the wall. the bottom of the plank is moving about twice as fast as the top

if the top of the plank were 2.5 meters up, the bottom would be 2.5 meters from the wall

then b' = aa'/b = a'= 0.22 m/sec, the top would then be moving at the same speed as the bottom. As the top goes higher up, the bottom moves faster. Initially at time t=0, the bottom was moving at zero m/sec. as the top approaches 5 meters up, the bottom's speed approaches an infinite number of meters per second. b' =-5(.22)/0 is undefined

Answer 2

Fun!

Let how high up on the building the plank is be represented by the variable, h.

Let the distance from the wall the plank is on the floor be represented by the variable, b.

Because the bottom of the plank is sliding on a path perpendicular to the floor, that tells us the plank is the hypotenuse of a right triangle with height, h, and base, b. This gives us a relationship between the length of the plank (which is 5), b, and h using the Pythagorean theorem.

h^2+b^2=5^2=25

From here we take derivatives with respect to time and get

2h (dh/dt) +2b(db/dt) =0

Now, you just need to plug in for h, dh/dt, and b then solve for db/dt. To find h you're going to need to use the h^2+b^2=25.

As an aside: My favorite part about this question is that as h approaches 5, we find that db/dt approaches infinity.