To solve this problem, you need to first address the geometry. Point P is a reference distance point with value equal to 0 km. The north-south road the car is driving north on is defined as the ordinate or y-axis. At time 0, the north-bound car is at y = 15 km (directly north of P) with a velocity or rate, dy/dt = 60 km/hr. The east-west road that the plane is flying east-bound 2 km directly above is defined as the abscissa or x-axis. At time t = 0, the east bound plane is at x = 10 km east of P and 2 km above at a velocity or rate, dx/dt = 190 km/hr. To define the distance between the points on the two roads at t = 0, where the car is on the y-axis and the plane directly above the x-axis, with a right angle at P, using Pythagorean's theorem or identity, c^2 = x^2 + y^2, were x is x-axis, y is y-axis, and c is the hypotenuse. However, we need to solve the rate of distance between the plane 2 km directly above the x-axis and the car on the y-axis. So we need another correlation to solve problem. There is another right angle between the point directly below the plane and the c hypotenuse. The axis to the plane is defined as the z-axis. The plane is flying parallel to the x-axis at z = 2 km (constant). Therefore, again using Pythagorean's theorem, with D defined as the hypotenuse or distance between the plane and car, z is the z-axis, and c is the lkower plane hypotenuse at a right angle to the z-axis.
Therefore, D^2 = c^2 + z^2, where we need to solve for dD/dt.
Substituting in for c^2 = x^2 + y^2 from first identity into second identity:
D^2 = (x^2 + y^2) + z^2
and taking derivative of each term with respect to t or time, therefore:
2*D*dD/dt = 2*x*dx/dt + 2*y*dy/dt + 0 (since z is constant)
divide by 2 on both sides,
D*dD/dt = x*dx/dt + y*dy/dt
Need to solve for D at t =0, x (at t = 0) = 10 km, y (at t = 0) = 15 km
at t =0,
D^2 = c^2 + z^2 = (x^2 + y^2) + z^2 = 10^2 + 15^2 + 2^2 = 100 + 225 + 4 = 329
D = sqrt(329)
Therefore solving for dD/dt, which is the distance rate between the car and plane at t = 0
dD/dt = (x*dx/dt + y*dy/dt)/D = (10*190 + 15*60)/sqrt(329) = (1900 + 900)/sqrt(329)
= 2800/sqrt(329) = 154.4 km/hr