Question

A jogger runs around a circular track of radius 70 ft. let (x,y) be her coordinates, where the origin is the center of the track. When the jogger's coordinates are (42,56), her x-coordinate is changing at a rate of 19 ft/s. Find dy/dt.

Answer 1

So we have a circle of radius 90 ft

by drawing a diagram you can see that she must be
going around the track in the clockwise direction

x = 70*cos(a*t)
y = 70*sin(a*t)

given at point (42, 56)
dx/dt = 19 ft/s

42 = 70*cos(a*t)
cos(a*t) = 42/70

56 = 70*sin(a*t)
sin(a*t) = 56/70

find dx/dt and dy/dt

dx/dt = -70*a*sin(a*t)
dy/dt = 70*a*cos(a*t)


19 = -70*a*sin(a*t)
19 = -70*a*56/70
a = -19/56

so dy/dt = 70*a*cos(a*t)
dy/dt = 70*(-19/56)*(42/70)
dy/dt = -19*42/56

dy/dt = -14.25