Docking a Boat A boat is pulled toward a dock by means of a rope wound on a drum that is located 5 ft above the bow of the boat. If the rope is being pulled in at the rate of 4 ft/s, how fast (in ft/s) is the boat approaching the dock when it is 27 ft from the dock? (Round your answer to one decimal place.)

4.1 ft/s is how fast the boat is approaching the dock

Explanation:

The illustration above forms a right angle triangle . Therefore Pythagorean theorem is applied

c² = a² + b²

where c = rope length to the drum

a = 5ft

b = distance of the boat to the drum

c² = 5² + b²

c² = 25 + b²

We take the derivative implicitly

2c dc/dt = 0 + 2b db/dt

2c dc/dt = 2b db/dt

c (dc/dt) = b (db/dt)

The rope length is decreasing so the rate is -4 ft/s , c = -4 ft/s

-4c/b =  db/dt

Checking how fast the boat is approaching the dock when it is 27 ft from the dock

c² = a² + b²

b = 27 ft

c² = 5² + 27²

c² = 25 + 729

c² = 754

c = √754

c = 27.4590604355

The rate the length changes is

db/dt =  -4c/b

db/dt = (-4 × 27.4590604355 )/b

db/dt =  -109.836241742 /27

db/dt  = - 4.06800895341

db/dt = - 4. 1 ft/s

4.1 ft/s is how fast the boat is approaching the dock

the displacement produced by the body.  

the amount of force subjected to the body.  

the angle between the direction of force and displacement