A plane, flying horizontally at an altitude of 1 km and a speed of 500 km/hr, passes directly over a radar station. find the rate in km/hr at which the distance from the plane to the station is changing when it is 2 km away from the radar station. Hint: this is not a max/min problem. into what category of problems does this problem belong?
This problem belongs to the category of related rates, which deals with finding how the rate of change of one quantity is related to the rate of change of another quantity. In this case, we want to find the rate at which the distance from the plane to the radar station is changing, given the plane's altitude and speed, and the distance from the plane to the radar station.
To solve this problem, we can use the Pythagorean theorem to relate the distance from the plane to the radar station with the plane's altitude.
Let's set up a coordinate system where the radar station is at the origin and the positive y-axis points vertically upwards. Let the plane's altitude be h and the distance from the plane to the radar station be d.
Using the Pythagorean theorem, we have:
d^2 = h^2 + (500t)^2
where t is the time in hours.
Now, we will differentiate both sides of the equation with respect to time to find the rates of change:
2d(d/dt) = 2h(dh/dt) + 2(500t)(500)(dt/dt)
Simplifying, we get:
d(d/dt) = h(dh/dt) + 500^2
To find the rate at which the distance from the plane to the radar station is changing when it is 2 km away from the radar station, we substitute d = 2 km = 2000 m into the equation.
Now, we need to find the rate of change of the plane's altitude, which is given as 1 km/hr = 1000 m/hr.
Finally, we substitute the known values into the equation and solve for the rate at which the distance is changing.
this is a related rates problem.
When the plane is x km (horizontally) away from the station, the distance y from the plane to the station is
y^2 = 1^2 + x^2
When x=2, y^2 = 5
So, we now know what we need.
2y dy/dt = 2x dx/dt
2√5 dy/dt = 2*2*500
dy/dt = 200√5 km/hr