Final answer:
To find the rate at which the angle of elevation is changing, we can use trigonometry and differentiation. By calculating the necessary values and substituting them into the differential equation, we find that the rate at which the angle of elevation is changing is approximately 0.0096 radians/second.
Step-by-step explanation:
To find the rate at which the angle of elevation is changing, we can use trigonometry. Let's denote the angle of elevation as θ and the height of the balloon as h. Since the balloon is rising vertically, the height h is directly related to the distance x between Madeleine and the balloon. We can use the tangent function to express this relationship:
tan(θ) = h / x
To find the rate at which θ is changing, we can differentiate both sides of the equation with respect to time:
sec^2(θ) dθ/dt = (dh/dt) / x
Since the balloon is rising vertically, dh/dt is given as 11 m/s. We are asked to find dθ/dt when h = 900 meters. To do this, we need to find x at that point. Using the Pythagorean theorem, we can write:
x^2 = 400^2 + 900^2
x ≈ 1006.25 meters
Substituting the values into the equation, we have:
sec^2(θ) dθ/dt = 11 / 1006.25
To find dθ/dt, we need to find sec(θ). Since tan(θ) = h / x, we have:
tan(θ) = 900 / 1006.25
θ ≈ 40.563°
Therefore, sec(θ) ≈ 1.267
Substituting this value into the equation, we can solve for dθ/dt:
1.267^2 dθ/dt = 11 / 1006.25
dθ/dt ≈ 0.0096 radians/second