# a balloon rises at a rate of 3 meters per second from a point on the ground 30 meters from an observer. find the rate of change of the angle of elevation of the balloon from the observer when the balloon is 30 meters above the ground...

so i drew a small picture depicting it but i'm still confused on what to do. am i basically finding the rate of change?? as in the DR/DT? i need someone to explain this to me, pleaseee! thank you.

## Oh, I see you're struggling with this problem. Don't worry, I'm here to help, or at least make you laugh in the process!

To tackle this problem, we need to find the rate of change of the angle of elevation of the balloon from the observer. This means we want to find the rate at which the angle is changing with respect to time. In mathematical terms, we need to find d(θ)/dt, where θ is the angle of elevation and t is time.

To start, let's create a right triangle with the observer, the balloon, and the ground. The height of the balloon is increasing at a rate of 3 meters per second, and when it is 30 meters above the ground, it means it has been ascending for 30/3 = 10 seconds.

Now, imagine the balloon as the hypotenuse of the right triangle, with the vertical distance being the height and the horizontal distance being the 30 meters from the observer. We want to find d(θ)/dt when the balloon is 30 meters above the ground.

To figure this out, we can use trigonometry. In our right triangle, the tangent of the angle of elevation (θ) is equal to the height of the balloon (30 meters) divided by the distance from the observer (30 meters). So, tan(θ) = 30/30, which simplifies to tan(θ) = 1.

Now, we can differentiate both sides of the equation with respect to time (t):

d(tan(θ))/dt = d(1)/dt

Applying the chain rule, we have:

sec^2(θ) * dθ/dt = 0

Since sec^2(θ) is never zero, we can divide both sides by sec^2(θ):

dθ/dt = 0

And there you have it – the rate of change of the angle of elevation of the balloon from the observer is 0. In other words, the angle of elevation remains constant at this particular instant when the balloon is 30 meters above the ground.

I hope this explanation helps, or at least brought a smile to your face!

## To solve this problem, you need to find the rate of change of the angle of elevation of the balloon with respect to time. Let's break it down step by step:

1. Let's assign some variables:
- Let "d" represent the distance from the observer to the balloon (30m in this case).
- Let "h" represent the height of the balloon above the ground.
- Let "θ" represent the angle of elevation of the balloon as observed by the observer.

2. The observer is 30 meters away from the balloon, so we have a right angle triangle formed by the observer, the ground, and the balloon. The side adjacent to the angle of elevation is the distance "d" and the side opposite is the height "h."

3. From trigonometry, we know that:
- The tangent of an angle is equal to the side opposite divided by the side adjacent.
- In this case, tan(θ) = h / d.

4. We want to find the rate of change of the angle of elevation, dθ/dt, when the height of the balloon is 30 meters. In other words, we want to find how fast the angle is changing at that specific height.

5. To get dθ/dt, we need to take the derivative of both sides of the tangent equation with respect to time (t):
- d/dt(tan(θ)) = d/dt(h / d).

6. Now, let's differentiate both sides:
- sec^2(θ) * dθ/dt = (d/dt(h) * d - h * d/dt(d)) / d^2.
- dθ/dt = (d/dt(h) * d - h * d/dt(d)) / (d^2 * sec^2(θ)).

7. Since we’re given the rate of change of height with respect to time (dh/dt) and the value of d (30m), we can substitute these values into the derivative equation and solve for dθ/dt when h = 30:
- dθ/dt = (dh/dt * d - h * 0) / (d^2 * sec^2(θ)).
- dθ/dt = (dh/dt * d) / (d^2 * sec^2(θ)).

8. With the given information that the balloon rises at a rate of 3 meters per second (dh/dt = 3 m/s) and d = 30m, we can substitute these values into the equation:
- dθ/dt = (3 * 30) / (30^2 * sec^2(θ)).

9. To find the sec^2(θ), we can use the Pythagorean theorem:
- d^2 = h^2 + d^2.
- 30^2 = h^2 + 30^2.
- h^2 = 30^2 - 30^2.
- h^2 = 900 - 900.
- h^2 = 0.
- h = 0.

10. Since h = 0, the balloon is still on the ground, and we cannot determine the angle of elevation when the balloon is on the ground. Therefore, the rate of change of the angle of elevation of the balloon from the observer cannot be determined at that height.

In conclusion, when the balloon is 30 meters above the ground, the rate of change of the angle of elevation of the balloon from the observer cannot be determined.

## call the angle of elevation α (alpha)

so what you want to find is dα/dt