Ira S. answered • 11/14/16

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So this involves right triangles. The height is always 100. Let the horizontal be x and the length of string be z.

So we have x

^{2}+ 100^{2}= z^{2}. Now take its derivative in terms of time to get2x(dx/dt) = 2z(dz/dt)

So at your specific moment z = 200, x = 100√3 and dx/dt = +8

substituting, that makes dz/dt = 800√3 / 200 or 4√3.

Part 2

sin a = 100/z = 100 z

^{-1 . }Now take the derivative in terms of t to getcos a (da./dt) = -100/ z

^{2 }(dz/dt)So we know z = 200, which makes this a 30-60-90 triangle, therefore a=30 degrees or π/6 radians.

Substitute to get

cos (π/6)(da/dt) = (-100/ 40000)(4√3)

√3 / 2 (da/dt) = -√3 / 100

da/dt = -1/50 radians

Hope this helped.