Question

A boat is pulled into a dock by means of a winch 12 feet above the deck of the boat. (a) The winch pulls in rope at a rate of 4 feet per second. Determine the speed of the boat when there is 15 feet of rope out.

Answer 1

Answer:

the speed of the boat is 6.67 ft/s

Step-by-step explanation:

Given;

height of the winch, h = 12 ft

the rate at which the winch pulls, the rope, = 4 ft/s

This form a right triangle problem;

let the height of the right triangle = h

let the base of the triangle = b (this corresponds to the horizontal displacement of the boat)

let the hypotenuse side = c

c² = b² + h²

$2c\frac{dc}{dt} = 2b\frac{db}{dt} + 2h \frac{dh}{dt}\\\\The \ height \ of \ the \ winch \ is \ not \ changing \\\\2c\frac{dc}{dt} = 2b\frac{db}{dt} + 2h (0)\\\\2c\frac{dc}{dt} = 2b\frac{db}{dt} \\\\c\frac{dc}{dt} = b\frac{db}{dt} ----(*) \\\\when;\\\\the\ hypotenuse \ c = 15 \ ft\\\\the \ the \ the \ height, h = 12 \ ft\\\\the \ base, b \ becomes ;\\\\b^2 = c^2 -h^2\\\\b^2 = 15^2 - 12^2\\\\b^2 = 81\\\\b = \sqrt{81} \\\\b = 9 \ ft\\\\\\from \ the \ equation (*) \ above;$

$c\frac{dc}{dt} = b \frac{db}{dt} \\\\dc/dt = 4 \ ft/s, \ \ c = 15 \ ft, \ \ b = 9 \ ft\\\\15 (4) = 9\frac{db}{dt} \\\\60 = 9 \frac{db}{dt} \\\\\frac{db}{dt} = \frac{60}{9} = 6.67 \ ft/s$

Therefore, the speed of the boat is 6.67 ft/s