.
s is the distance between you and the plane.
y=3 is the altitude of the plane.
x=4 is the distance between you and the point on the ground.
The Pythagorean formula gives us:
s^2=x^2+y^2 (color(red)(Equation- 1))
We can plug the values of x and y in and find s
s^2=4^2+3^2=16+9=25
s=5 miles
Now, we take the derivative of both sides of Equation 1with respect to time (t):
2s((ds)/dt)=2x(dx/dt)+2y(dy/dt)
(ds)/dt=(2x(dx/dt)+2y(dy/dt))/(2s)=(2(x(dx/dt)+y(dy/dt)))/(2s)=
(cancelcolor(red)2(x(dx/dt)+y(dy/dt)))/(cancelcolor(red)2s)=
(ds)/dt=(x(dx/dt)+y(dy/dt))/s
At the moment when the point on the ground is 4 miles away from you, x is changing at a rate of 500 miles per hour. This means:
dx/dt=500
Since the altitude is not changing,:
dy/dt=0
Now, we can plug these values in and calculate (ds)/dt which is the rate of change of the distance between you and the plane at that moment:
(ds)/dt=((4)(500)+(3)(0))/5=(2000+0)/5=2000/5=400 miles per hour
