Difficult Rates of Change Questions (1 Viewer)

[xokki]

Hi I was wondering if anyone could help me with these questions

1) a pigeon at an altitude of 100m passes directly over a loft. If the pigeon is flying horizontally at 5m/sec, find the rate at which the pigeon is approaching
the loft when it is 75m away. The answer for this is 3m/s

2) A heavy object is attached to a 30m long rope which passes over a pulley 12m above the ground. The other end of the rope is held in a man's hands at a distance of 1.5m
above the round. If the man is walking away at a rate of 2m/sec, find how fast the object is being raised when it is 3m above the ground. Ans is 1/6pi m/sec

 

[hayabusaboston]

Hi I was wondering if anyone could help me with these questions

1) a pigeon at an altitude of 100m passes directly over a loft. If the pigeon is flying horizontally at 5m/sec, find the rate at which the pigeon is approaching
the loft when it is 75m away. The answer for this is 3m/s

2) A heavy object is attached to a 30m long rope which passes over a pulley 12m above the ground. The other end of the rope is held in a man's hands at a distance of 1.5m
above the round. If the man is walking away at a rate of 2m/sec, find how fast the object is being raised when it is 3m above the ground. Ans is 1/6pi m/sec

Draw up triangle, let vertical bit be 100, horizontal bit be x, hypotenuse be m.

Now dx/dt is given as 5m/s, and we essentially want the rate of change of the distance of m, so we want dm/dt. Using the rule of rates,

dm/dt=dx/dt.dm/dx

to get dm/dx. we need a relationship with m and x. Since its a triangle, m^2=x^2+100^2

Thus m=sqrt(x^2+100^2)

so we get dm/dx now by differentiating. You get a final result of dm/dx=x/sqrt(x^2+100^2)

Times it by dx/dt which is 5, then you get 5x/sqrt(x^2+100^2)

sub in x=75 and you get your answer of 3.

Unless I visualised an incorrect picture, question 2 appears to be just about the same as question 1 but with different numbers?

 

[xokki]

cool, thanks for answering the first one, can anyone help with the second, having trouble visualising the question :L

 

[hayabusaboston]

U sure you worded the 2nd one correctly? lol it doesnt seem to make sense or maybe im just not thinking properly haha

 

[hayabusaboston]

OH WAIT of course lol I just got the picture haha. Working it out and will post in a sec

Hmm wait, is it supposed to be like, essentially 2 triangles with a common height, with the hypotenuse of one getting shorter as the hypotenuse of another getting longer?

 

[xokki]

im not sure, but i visualised it as a trapezium sort of thing ><