Hello. I was wondering if anyone could provide some insight into how to solve the following Calculus word problem:
Max is walking his dog Beau in the Cartesian plane, with the leash between them at its full 10 ft extension.
They walk up the negative y-axis, but just as Beau reaches the origin he spots a squirrel and starts running along the line y = x (in the positive direction), dragging Max behind him. The leash stays at full extension throughout and at any given instant is tangent to the curve Max is dragged along.
If (x, y) is a point on the curve Max is dragged along, find dy/dx as a function of x.
Hint: If Beau is at (u, u) when Max is at (x, y), then the slope of the line joining them and the distance between them, respectively, are?
Any help would be greatly appreciated.
The differential equation of the trajectory is $$y'=\frac {y-u}{x-u}$$with $u$ obtained solving$$(x-u)^2+(y-u)^2=100$$and choosing the root with the plus sign.
The initial condition is $\,y(0)=-10\,$ of course.