The angle is decreasing at a rate of 0.0133 radians per second.
The height of the kite from the ground is y = 75 feet.
The kite moves horizontally at a speed of dx/dt = 4 ft/s
The length of the string L = 150 feet
The formula for the sine angle is:
sin ( θ ) = 75/150
sin ( θ ) = 1/2
The horizontal displacement (x) is calculated using the following tangent ratio:
tan θ = 75/x
1 / tanθ = x/75
cot θ = x/75
Differentiating the above equation with respect to t.
- csc² ( θ ) × dθ/dt = 1/75 × dx/dt
Substituting the values in the equation,
- csc² ( θ ) × dθ/dt = 1/75 × 4
- csc² ( θ ) × dθ/dt = 4/75
Now,
csc θ = 1/sin θ = 2
So,
- 2² × dθ/dt = 4/75
dθ/dt = - 1/75
dθ/dt = 0.0133
Hence, the angle is decreasing at a rate of 0.0133 radians per second.
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