Question

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.)

Answer 1

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

Step-by-step 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