A kite 100 ft above the ground moves horizontally at

Painevg 2021-12-18 Answered
A kite 100 ft above the ground moves horizontally at a speed of 8 ft/s. At what rate is the angle between the string and the horizontal decreasing when 200 ft of string has been let out?
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Expert Answer

Lindsey Gamble
Answered 2021-12-19 Author has 38 answers
At the moment, dxdt=8 fts and we want to find dθdt. We can find the curren angle first
θ=arcsin100200=π6
Get an equation to relate the x and θ. Differentiate with respect to time t
x100=cotθ
0.01dxdt=csc2θdθdt
dθdt=0.01dxdtcsc2θ
=0.01(8)csc2π6 plug in known values
=0.08(2)2
=0.02 rads
The negative means it is decreasing at 0.02 radians per second (1.15s)

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Chanell Sanborn
Answered 2021-12-20 Author has 41 answers
The angle between the horizontal and the kite θ
Let Height of kite be h
Horizontal distance of kite from where the string is held be x
Then the length of the string be s and will be given by the pythagoras theorem as follows:
s2=h2+x2
Differentiate throughout to get
2sdsdt=2hdhdt+2xdxdt
2sdsdt=2×(0)+2x×8
sdsdt=8x(1)

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