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.

Question

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 0.30 meter per second. How fast is the end of the plank sliding along the ground when it is 2.5 meters from the wall of the building?

Answer 1

Let

t be a time variable,

t1 be the time instant when the end of the plank sliding along the ground is 2.5 meters from the wall of the building,

h(t) be the height of the end of the plank sliding along the wall,

h'(t) = 0.3 m/s be the speed of the end of the plank sliding along the wall,

d(t) be the distance from the wall of the end of the plank sliding along the the ground,

d(t1) = 2.5m be distance at which we are to evaluate the speed,

l = 5m be the length of the plank.

Since we have a right triangle

h² + d² = l²

h(t1) = √(l² - d²(t1)) = √(5² - 2.5²) = 4.33 (approximately)

We calculate the sought speed by expressing d(t) and getting its derivative.

d = (l² - h²)^1/2

d' = (1/2)(l² - h²)^-1/2 (-2hh') = -hh'/d

d'(t1) = -4.33×0.3/2.5 = -0.52 m/s (approximately)

At time t1 the distance from the wall will be decreasing at the rate of about 0.52 m/s.