an airplane is flying at an altitude of 6.7 miles towards a point directly over an observer. if the speed of the plane is 499 miles per hour, find the rate at which the angle of observation, Ɵ, changing by at the moment when the angle is 21 degrees.
As usual, draw a diagram. If the plane is x miles away horizontally,
x/6.7 = tan Ɵ
dx/6.7 = sec^2 Ɵ dƟ
-499/6.7 = sec^2(21°) dƟ
dƟ = -65°/hr
To find the rate at which the angle of observation (θ) is changing, we can use related rates.
Let's start by drawing a diagram to visualize the situation.
```
/|\
/ | \ Plane
H / | \
/ | \
/ | \
/_____|_____\
O
```
In the diagram above, O represents the observer on the ground, and H represents the airplane at a height of 6.7 miles. The angle of observation is denoted by θ.
To find the rate at which θ is changing, we need to relate it to the given variables: the height (H) and the rate of change of the distance between the observer and the airplane.
We know the speed of the airplane is 499 miles per hour. Therefore, the rate at which the distance between the observer and the airplane is changing can be calculated by taking the derivative of the distance with respect to time (dH/dt).
Now let's determine the relationship between the height (H), the distance between the observer and the airplane (D), and the angle of observation (θ) using trigonometry.
We can see that tan(θ) = H/D, where H is the height and D is the distance between the observer and the airplane.
Rearranging the equation, we have D = H / tan(θ).
Differentiating both sides with respect to time (t), we get dD/dt = (dH/dt * tan(θ) - H * sec^2(θ) * dθ/dt).
Since we want to find dθ/dt, we can rearrange the equation to solve for it:
dθ/dt = (dD/dt + (H * sec^2(θ) * dθ/dt)) / (H * tan(θ))
Now we can substitute the given values into the equation. We have H = 6.7 miles, dD/dt = 499 mph, and θ = 21 degrees.
Using these values, we can calculate the rate at which the angle of observation is changing.