Related Rates problem

ashshea

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Oct 12, 2014
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I cannot for the life of me figure out this problem:

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 20 feet above the deck of the boat (see figure). The winch pulls in rope at a rate of 2 feet per second. Find the acceleration of the boat when there is a total of 25 feet of rope out. (Round your answer to three decimal places.)


So how I've been going about it is
s(hypotenuse)=25 feet
y=20 feet (constant)
(ds/dt)=2 ft/sec

I found x using the pythagorean theorem which makes x=15 feet
So next I found the derivative of the pythagorean theorem using implicit differentiation>>> x^2+20^2=s^2 which becomes 2x(dx/dt)=2s(ds/dt)
Then I substituted in everything>>> 2(15)(dx/dt)=2(25)(2) which becomes (dx/dt)=100/30 or 3.33 ft/sec^2 which is incorrect but I have no idea why?? Help anybody? I have one more shot at the right answer and I just don't know what I'm doing wrong.
 
I cannot for the life of me figure out this problem:

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 20 feet above the deck of the boat (see figure). The winch pulls in rope at a rate of 2 feet per second. Find the acceleration of the boat when there is a total of 25 feet of rope out. (Round your answer to three decimal places.)


So how I've been going about it is
s(hypotenuse)=25 feet
y=20 feet (constant)
(ds/dt)=2 ft/sec

I found x using the pythagorean theorem which makes x=15 feet
So next I found the derivative of the pythagorean theorem using implicit differentiation>>> x^2+20^2=s^2 which becomes 2x(dx/dt)=2s(ds/dt)
Then I substituted in everything>>> 2(15)(dx/dt)=2(25)(2) which becomes (dx/dt)=100/30 or 3.33 ft/sec^2 which is incorrect but I have no idea why?? Help anybody? I have one more shot at the right answer and I just don't know what I'm doing wrong.

First, consider the "hint" given in the problem. I would restate that as, "Constant velocity means zero acceleration."

Second, you've stated that " (dx/dt)=100/30 or 3.33 ft/sec^2 ". Please track your units throughout the math steps and verify that you actually end up with ft/sec^2 (rather than simply assuming that those are the correct units).

Now ask yourself, "What have I actually calculated? Was it acceleration or velocity?"
 
Or, possibly another way depending on what you can use:
a = acceleration
v = velocity
s = distance
a = dv/dt = d(ds/dt)/dt
You are given
ds/dt = 2
so what is d(ds/dt)/dt.
 
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