A drone aeroplane is flying horizontally to a constant height of 4000 ft above a fixed radar tracking station. At a certain instant the angle of elevation theta is 30 degrees and decreasing, and the speed of the aeroplane is 300 mi/h
a) How fast is theta decreasing at this instant? express results in units of deg/s
b) How fast is the distance between the aeroplane and the radar station changing at this instant? express the rates in units of ft/s 1 mi = 5280ft
To solve this problem, we need to use trigonometry and related rates. Let's start by drawing a diagram to visualize the scenario:
```
A (drone)
/ |
/ |
d / | h
i / |
s / |
t / |
/ |
R (radar station)
```
Here, A represents the drone, R represents the radar station, d represents the horizontal distance between the drone and the radar station, h represents the drone's height above the radar station, and θ represents the angle of elevation.
Given information:
- Height of the drone, h = 4000 ft
- Speed of the drone, v = 300 mi/h = (300 * 5280) ft/h
- Angle of elevation, θ = 30 degrees
a) To find how fast θ is decreasing, we need to differentiate θ with respect to time t. Let's assume θ is a function of time, θ(t). We want to find dθ/dt when θ = 30 degrees.
Since the drone is flying horizontally and maintains a constant height, we have a right-angled triangle formed by the drone, the radar station, and the distance between them (d). Using trigonometry, we know that:
sin(θ) = h/d
Differentiating both sides of this equation with respect to time t:
d(sin(θ))/dt = (d/dt)(h/d)
Using the chain rule, we can rewrite this as:
cos(θ) * dθ/dt = (dh/dt) / d
We want to find dθ/dt, so let's isolate it:
dθ/dt = (dh/dt) / (d * cos(θ))
Now we need to find dh/dt and d. We are given that dh/dt = 0, as the height of the drone is constant. And since the drone is flying horizontally, d/dt = v.
Now we can substitute the given values into the equation:
dθ/dt = 0 / (d * cos(θ))
Since we know that h = 4000 ft and θ = 30 degrees, we can find d using trigonometry:
sin(θ) = h/d
sin(30) = 4000/d
d = 4000 / sin(30)
Hence, we can find dθ/dt:
dθ/dt = 0 / [(4000 / sin(30)) * cos(30)]
b) To find how fast the distance d is changing, we need to differentiate d with respect to time t. Let's assume d is a function of time, d(t). We want to find dd/dt when θ = 30 degrees.
Using the Pythagorean theorem, we know that:
d^2 = h^2 + d^2
Differentiating both sides of this equation with respect to time t:
2d * dd/dt = 2h * dh/dt
Again, we know that dh/dt = 0, so the equation simplifies to:
2d * dd/dt = 0
Dividing both sides by 2d:
dd/dt = 0 / (2d)
Now we can substitute the given values into the equation:
dd/dt = 0 / (2 * [(4000 / sin(30)) * cos(30)])
Therefore, dd/dt = 0.
So, at the given instant when θ = 30 degrees:
a) θ is decreasing at a rate of dθ/dt = 0.
b) The distance between the drone and the radar station is not changing, dd/dt = 0.
If I get the picture accurately
a) height/horizontaldistance away= sinTheta
height= distancaway*sinTheta
take the deriviative
0=d (distance)/dt *sinTheta- distance*cosTheta dTheta/dt
well d (distance)/dt=velocity= 300mi/hr
dTheta/dt= tanTheta*300mi/hr / distance
but distance= height/sin30 then solve for dTheta/dt. Change mi/hr to feet/sec (ie 60mph=88ft/second)