Answer:
Explanation:
Let's consider the situation described in the problem:
The balloon rises at a rate of 6 meters per second.
The balloon is initially 10 meters from an observer on the ground.
We want to find the rate of change of the angle of elevation of the balloon from the observer when the balloon is 10 meters above the ground.
We can use trigonometry to relate the angle of elevation to the distance between the observer and the balloon. Let's call the angle of elevation θ and the distance between the observer and the balloon d. Then we have:
tan(θ) = (height of balloon) / d
Differentiating both sides with respect to time, we get:
sec^2(θ) * dθ/dt = (dh/dt) / d
where h is the height of the balloon above the ground. We are given that (dh/dt) = 6 m/s when h = 10 m, and we want to find dθ/dt when h = 20 m. Since the observer is 10 meters away from the balloon, we have d = 10 m when h = 20 m.
Plugging in the values we have, we get:
sec^2(θ) * dθ/dt = (6 m/s) / 10 m
We need to find sec^2(θ) when h = 20 m. Let's call this value s:
s = sec^2(θ) = (d^2 + h^2) / d^2 = (10^2 + 20^2) / 10^2 = 5
Plugging this into the equation above, we get:
5 * dθ/dt = 6 m/s / 10 m
Solving for dθ/dt, we get:
dθ/dt = (6 m/s / 10 m) / 5 = 0.12 rad/s
Therefore, the rate of change of the angle of elevation of the balloon from the observer when the balloon is 10 meters above the ground is 0.12 rad/s.