Related Rates

[Ezalia]

Could you guys help me with this problem? I genuinely do not know how to do it (cuz im stupid) but I'm very lost. It would be great if you could show your work but you don't have to. Thank you!

Find the acceleration of the specified object. (Hint: Recall that if a variable is changing at a constant rate, its acceleration is zero.)
A boat is pulled into a dock by means of a winch 15 feet above the deck of the boat (see figure). The winch pulls in rope at a rate of 7 feet per second. Find the acceleration (in ft/sec2) of the boat when there is a total of 17 feet of rope out. (Round your answer to three decimal places.)

 

[R.M.]

This is what I teach my students: Find __________ when __________. The first blank is the rate you are trying to find. The second blank is your initial conditions. Fill those in first, and you'll be on your way.

 

[HallsofIvy]

At any given time the rope is the hypotenuse of a right triangle where one leg is the distance of the boat from the dock and the other is the height of the winch above the boats deck. Though, oddly, it isn't labeled in the picture, call the distance the boat is from the dock "x". Then, by the Pythagorean theorem, \(\displaystyle z^2= x^2+ y^2\).

Differentiate each part of that with respect to the time, t: \(\displaystyle 2z\frac{dz}{dt}= 2x\frac{dx}{dt}+ 2y\frac{dy}{dt}\). The problem tells us that y, the height of the winch above the deck of the boat, is a constant, 15 feet. So what is \(\displaystyle \frac{dy}{dt}\)? We are told that the rope is being drawn in at 7 feet per second. So what is \(\displaystyle \frac{dz}{dt}\)? Given that y is the constant 15 feet and that, at the moment asked about, z= 17 feet, what is x at that moment?

You now have enough information to determine dx/dt as a function of t. The acceleration is the derivative of that. (Asking for the acceleration rather than the speed is a little odd but it's really the same thing.)