how fast in ft/sec is the boat approaching the dock when it is 34 ft from the dock?

Question

A boat is pulled toward a dock by means of a rope wound on a drum that is located 2 ft above the bow of the boat. If the rope is being pulled in at the rate of 3 ft/sec, how fast is the boat approaching the dock when it is 34 ft from the dock? (Round your answer to one decimal place.)

Answer 1

I half agree with the previous answer, but in the end the result is wrong.

correction, I had ratio inverted

if the speed toward the dock is considered negative, then dx/dt= [-3*√(34²+2²)]/34

which is about -3.005 feet/sec.

it is from the calculus 2x*dx/dt=2h*dh/dt where h is hypotenuse

Answer 2

Sketch a right triangle with vertices at the front of the boat, the point on the dock directly below the drum, and the drum

Let x: the length of the horizontal leg of the triangle (the distance from the boat to the dock)

h: the length of the hypotenuse (the length of the towrope)

x² + 2² = h²

2x·dx/dt = 2h·dh/dt

34·dx/dt = √(34²+4)·(- 3)

dx/dt = - 3√(34²+ 4) / 34 ft/sec