A point is moving along the circle in such a way that its x -coordinate is decreasing at a rate of 3 units per second. At what rate is the y -coordinate decreasing at the instant when the point has reached ( 2, 2)?
A)8 units per second
B)6 units per second
C)10 units per second
D)7 units per second
E)5.5 units per second

Question

A point is moving along the circle  in such a way that its x -coordinate is decreasing at a rate of 3 units per second. At what rate is the y -coordinate decreasing at the instant when the point has reached ( 2, 2)?
A)8 units per second 
B)6 units per second 
C)10 units per second 
D)7 units per second 
E)5.5 units per second

Quizplus · Accepted Answer

Let the point be P and let O be the center of the circle. Let x and y be the x and y-coordinates of P respectively. Also, let r be the radius of the circle.
At any instant, we have x^2+y^2=r^2 by the Pythagorean theorem. Differentiating both sides with respect to time t, we get:
2x(dx/dt) + 2y(dy/dt) = 0
At the instant when P is at (2,2), we have x=2 and y=2, and dx/dt=-3 (since x is decreasing at a rate of 3 units per second). Solving for dy/dt, we get:
dy/dt = (-x/y)(dx/dt) = (-2/2)(-3) = 3
Therefore, at that instant, the y-coordinate of P is decreasing at a rate of 3 units per second. The answer is B.

A Point Is Moving Along the Circle in Such