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May 4th, 2009, 02:08 AM   #1
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Related rates

Please help me to check the solutions. Thanks for your help

1. a particle moves along a path described by y=x^2. at what point along the curve are x and y changing at the same rate? Find this rate if at that time t we have x=sin(t) and y= (sin(t))^2
Answer :
dy/dx = 2x = 1 => x = 1/2 and y = 1/4
Point is (1/2, 1/4).
dy/dt = 2sint cost and dx/dt = cost
dy/dt = dx/dt => 2sint cost = cost => sint = 1/2 => t = ?/6
=> rate is cost = cos(?/6) = ?3/2.

2. A man 6 ft tall walks at rate of 200 feet per minute towards a street light which is 18 ft above the ground. At what rate is the tip of his shadow moving?
Answer :
let shadow of street light = h and dh/dt = rate of change in street light shadow
let shadow of man to the street light= r and rate of change of the tip of his shadow = dr/dt

The man and the street light form two similar triangles which gives:
height of street light : height of man = shadow of street light : shadow of man
6 : 18 = r-h : r
1:3 = r-h : r
r=3r-3h
2r=3h
differentiate each with respect to time t
2dr/dt = 3dh/t
2dr/dt = 3 (-200)
dr/dt = -300 ft/minute (I'm not sure whether it should be -300 or +300)
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May 4th, 2009, 02:31 PM   #2
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Re: Related rates

#2:

Using similar triangles. Let x=distance between man and pole and y=length of his shadow.



Solving for y:



We are told dx/dt=200

 
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