Question

A jogger runs around a circular track of radius 45 ft. Let (x,y) be her coordinates, where the origin is the center of the track. When the jogger's coordinates are (27, 36), her x-coordinate is changing at a rate of 20 ft/s. Find dy/dt.

Answer 1

Answer:

dy/dt =  -5/9  ft/s

Step-by-step explanation:

The track can be expressed as a circle or radius 45ft as follows:

$x^{2} +y^{2} =45^{2} \\\\\\x^{2} +y^{2} =2025$

Implicit derivate for getting dy/dt

$y^{2} =2025-x^{2} \\\\2y\frac{dy}{dt} =0-2x\frac{dx}{dt}$

Then:

$\frac{dy}{dt} =\frac{-2x\frac{dx}{dt} }{2y} \\\\\frac{dy}{dt} =\frac{-x}{y} \frac{dx}{dt}$

Solving using the conditions. when x= 27 and y= 36 ; dx/dt=20  we have:

$\frac{dy}{dt} =\frac{-27}{36}(20)=-15 ft/s$