Related Rates: Kite String Angle Change

[Brandon_R]

Homework Statement

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?

Homework Equations

tan \o = x/y

The Attempt at a Solution

After solving for the sides of the triangle i got X = 173.21, Y = 100 and Z = 200 and \o = 1.047 rads

then i took the derivative of the equation that links \o with the opposite and adjacent of the triangle and plugged in the variables which gave me:

I got d\o/dt = .055 rad/sec

 

[rl.bhat]

String mαkes an angle φ with horizontal.
Hence tanφ = y/x, where y is constant and x is variable.
When the string in overhead, the length of the string is 100 ft. 200 ft of string is let out. So the final length of the string is 300 ft.
Taking the derivative on both the side, with respect to time, you get
(-y/x^2)(dx/dt) = sec^2(φ)(dφ/dt)...(1)
sec^2(φ) = 1 + tan^2(φ) = 1 + (y/x)^2
L = 300 ft, y = 100 ft. find x. dx/dt is given. Substitute these values in eq.(1) and find d(φ)/dt