Related Rates

[quaint sedge]

man its so tiny

💀

the problem: A kite 100 ft above the ground moves horizontally at a speed of 7 ft/s. At what rate (in rad/s) is the angle (in radians) between the string and the horizontal decreasing when 200 ft of string have been let out?

i'm calling the horizontal x, and the string y. the other side is 100 of course

the relationship is theta = tan(100/x)

I take the derivative, i get:
d[theta]/dt = 1/((100/x)^2 + 1) * d/dt[100/x]
d[theta]/dt = 1/((100/x)^2 + 1) * (-100x^(-2) * dx/dt)

when y is 200, x is 100sqrt3 using the pythagorean theorem

so i plug in x = 100sqrt3, and dx/dt = 7

I get 1/((sqrt3/3)^2 + 1) * (-100(100sqrt3)^-2) * 7) = -7/400 rad/sec or -0.0175 rad/sec