A plane that is flying horizontally at an altitude of 6 kilometers and a speed of 570 kilometers per hour passes directly over a radar station. How fast is the distance between the plane and the radar station increasing when the distance between the two is 14 kilometers
If the desired distance is z when the plane has moved x beyond the flyover point,
x^2+36=z^2
2x dx/dt = 2z dz/dt
Now just find x when z=14, and plug in the numbers to find dz/dt.
To find how fast the distance between the plane and the radar station is increasing, we need to use the concept of related rates.
Let's call the distance between the plane and the radar station at any time t as "d" (in kilometers). We are given that the plane is flying horizontally at a constant speed of 570 kilometers per hour and that its altitude is held at a constant height of 6 kilometers.
We want to find the rate at which d is changing with respect to time t when the distance between the two is 14 kilometers. To find this rate, let's differentiate the equation d^2 = (altitude)^2 + (horizontal distance)^2 with respect to time t.
Differentiating both sides of the equation, we get:
2d * (dd/dt) = 2(altitude) * (d(altitude)/dt) + 2(horizontal distance) * (d(horizontal distance)/dt)
Since the plane is flying horizontally, the change in altitude (d(altitude)/dt) is 0. Therefore, the equation simplifies to:
2d * (dd/dt) = 2(horizontal distance) * (d(horizontal distance)/dt)
Now, we can substitute the given values into the equation:
2 * 14 * (dd/dt) = 2 * (horizontal distance) * (d(horizontal distance)/dt)
Simplifying further:
28 * (dd/dt) = 2 * 14 * (d(horizontal distance)/dt)
Since the plane is flying horizontally at a constant speed of 570 kilometers per hour, the horizontal distance is changing at a constant rate, which is equal to the speed of the plane. So, (d(horizontal distance)/dt) = 570 km/h.
Substituting this value into the equation:
28 * (dd/dt) = 2 * 14 * 570
Simplifying:
28 * (dd/dt) = 2 * 7980
Now, we can solve for (dd/dt), which is the rate at which the distance between the plane and the radar station is changing.
28 * (dd/dt) = 15960
(dd/dt) = 15960 / 28
(dd/dt) ≈ 570 km/h
Therefore, when the distance between the plane and the radar station is 14 kilometers, the rate at which this distance is increasing is approximately 570 kilometers per hour.