A hot air balloon rising vertically is tracked by an observer located 4 miles from the life off point. At a certain moment, the angles between the observers line of sight and the horizontal is Pi/4 and it is changing at a rate of 0.1. How fast is the balloon rising at this moment
To find the rate at which the balloon is rising, we need to use trigonometry and differentiation.
Let's denote the height of the balloon as h (in miles) and the angle between the observer's line of sight and the horizontal as θ (in radians). We are given that θ = π/4.
First, let's draw a diagram to visualize the situation:
```
C
/|
/ |
h / |
/ |
/ | x
A / |
/ θ |
/______B
4 miles
```
Here, A is the observer, B is the liftoff point of the balloon, and C is the current position of the balloon. We know that AB = 4 miles, and θ = π/4.
Using trigonometry, we can write the following relationship:
tan(θ) = h / x
Taking the derivative of both sides with respect to time (since we want to find the rate of change of height with respect to time), we get:
sec^2(θ) * dθ/dt = dh/dt / x
Now, we need to find dθ/dt. It is given that dθ/dt = 0.1 radians per unit of time.
Plugging in the given values, we have:
sec^2(π/4) * 0.1 = dh/dt / 4
Simplifying, we find:
(2/√2)^2 * 0.1 = dh/dt / 4
2 * 0.1 = dh/dt / 4
0.2 = dh/dt / 4
Cross multiplying, we get:
dh/dt = 0.2 * 4
dh/dt = 0.8 miles per unit of time
Therefore, the balloon is rising at a rate of 0.8 miles per unit of time at the given moment.
At height h,
h/4 = tanθ
1/4 dh/dt = sec^2θ dθ/dt
So, plug in your numbers and find dh/dt