Question 1163077
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According to the condition, you have a right angled triangle.


One its leg is vertical and has constant dimension  v = (3.05 - 0.61) = 2.44 meters.


The other leg is horizontal and its dimension h increases with the rate of 


    {{{(dh)/(dt)}}} = 4 km/h = 4000/3600 m/s = {{{10/9)}}} m/s.


The hypotenuse  "c"  has the length  c = {{{sqrt(v^2 + h^2)}}}  and it is the length of the rope between the pulley and 
the car's rear bumper.


The value under the question is  the derivative of "c"  over time "t"


    {{{(dc)/(dt)}}} = (2*h*h'(t))/sqrt*(v^2 + h^2) = (2*h*h'(t))/sqrt*(v^2 + h^2).


You substitute the given data into the formula  and calculate 


    {{{(dc)/(dt)}}} = {{{(2*2.44*(10/9))/sqrt(2.44^2 + 4.88^2)}}} = 0.993808 m/s.


<U>ANSWER</U>.  The rope is going through the pulley at the rate of 0.993808 m/s, under given conditions.
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Solved.


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The major lesson to learn from my post is THIS:


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    After reading the post. you should ask yourself:

        What is given and what they want to get from me ?



    In this problem, they want you find the derivative of the length of the hypotenuse over the time.


    As soon as you understood it, the rest is just a technique.
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