kite problem for calculus

Question

A child is flying a kite attached to a 20 m long string. At first, the kite is directly above the child, but it blows east. The child notices that the angle between the string and the ground is decreasing by 5° each second. How quickly is the kite moving horizontally when the angle between the string and the ground is 76°?

Answer 1

This forms a right triangle with a hypotenuse of 20 m.

Let x be the horizontal distance.

dθ/dt = -5°/sec

x/20 = cos(θ)

(dx/dt)/20 = -sin(θ) dθ/dt

dx/dt = -20sin(θ) dθ/dt = -20sin(76°)(-5) ≈ 97 m/sec