Answer: Let's set up a diagram and define the variables to represent the varying quantities:
Let:
H = Height of the hot air balloon above the ground (in feet),
D = Distance from you to the hot air balloon (in feet),
θ = Angle of elevation from you to the hot air balloon (in radians).
Initially, when the balloon is at the ground, we have:
H = 0 feet (since it just lifted off),
D = 500 feet (the initial distance between you and the balloon).
As the balloon rises, H and D will change with time. When the balloon is at a certain height H above the ground, and you are still looking at it with an angle of elevation θ, we want to find how fast the angle θ is changing.
Since the angle of elevation is the angle between the line of sight (your gaze) and the horizontal plane, we have a right triangle formed between you, the hot air balloon, and a point on the ground directly beneath the balloon. Let's label the right triangle with sides and angles:
/
/ H
/ |
/ |
/ |θ
/______|
D
We have the following trigonometric relationship:
tan(θ) = H / D.
Differentiate both sides with respect to time t:
d(tan(θ)) / dt = d(H / D) / dt.
Using the chain rule, we have:
sec^2(θ) * dθ / dt = (dH / dt * D - H * dD / dt) / D^2.
Now, we need to find expressions for dH/dt (the rate at which the height of the balloon is changing) and dD/dt (the rate at which the distance from you to the balloon is changing).
Given:
dH/dt = 2 ft/s (the rate at which the balloon rises),
D = 500 ft (the distance between you and the balloon remains constant).
We can also find dD/dt using the Pythagorean theorem. When the balloon is at height H, the distance from you to the balloon is given by:
D^2 = H^2 + 500^2.
Differentiate both sides with respect to time t:
2D * dD/dt = 2H * dH/dt.
Now, solve for dD/dt:
dD/dt = (H * dH/dt) / D.
Now, plug the values into the expression we derived earlier:
sec^2(θ) * dθ / dt = (2 * D - H * (H * dH/dt) / D) / D^2.
At the given heights H = 75 ft and H = 150 ft, we can calculate the corresponding angles of elevation θ using the relation tan(θ) = H / D:
When H = 75 ft:
θ = arctan(75/500) ≈ 8.13 degrees.
When H = 150 ft:
θ = arctan(150/500) ≈ 16.26 degrees.
Now, substitute the corresponding values of θ, H, and dH/dt into the expression:
When H = 75 ft:
sec^2(θ) * dθ / dt ≈ (2 * 500 - 75 * (75 * 2)) / (500^2) ≈ -0.0040 rad/s ≈ -0.229 degrees/s.
When H = 150 ft:
sec^2(θ) * dθ / dt ≈ (2 * 500 - 150 * (150 * 2)) / (500^2) ≈ -0.0160 rad/s ≈ -0.918 degrees/s.
So, the rate at which your angle of elevation to the balloon is changing is approximately -0.229 degrees per second when the balloon is 75 feet above the ground and approximately -0.918 degrees per second when the balloon is 150 feet above the ground. The negative sign indicates that your angle of elevation is decreasing as the balloon rises.