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rates of change??plzz help!?
A man standing 7 feet from the base of a lamppost casts a shadow 10 feet long. If the man is 6 feet tall and walks away from the lamppost at a speed of 150 feet per minute, at what rate, in feet per minute, will his shadow lengthen?
Draw a right angled triangle ABC. B is the right angle. The vertical AB is the man. The horizontal BC is his shadow. Extend the hypotenuse AC at A to some length, stop it at point P. Make a right angled triangle from PQB. Where Q is the right angle. Note PQ parallel to AB. PQ is the lamp post. The triangles PQC and ABC are similar triangles. Let height of pole be h.
Then
h/17=6/10
h=10.2 feet.
Let length of shadow be L, let the distance of man from base of post be q. By the similar triangles,
10.2/(q+L)=6/L
Solve for L,
L= (10/7) q.
Given dq/dt = 150 feet per minute, we have to find dL/dt
dL/dt = dL/dq * dq/dt (by chain rule)
dL/dt = 10/7 * 150 = 1500/7 feet/second.
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