How fast is it’s distance from the origin changing

Question

An ant is walking left along the curve y=2/x. It’s x-coordinate is changing at 1 unit per minute. When the ant is at point (2,1), how fast is it’s distance from the origin changing?

Answer 1

The point on the curve is given by (x, 2/x). Since the ant is moving left to right the rate of change of x with respect to t is actually -1 unit per minute.

The distance from the origin to a general point is:

z² = x² + (2/x)² = x² +4x^-2

Dimensions are changing with respect to time, so differentiate with respect to t:

2z (dz/dt) = 2x (dx/dt) - 8x^-3(dx/dt)

dz/dt = [x(dx/dt) - 4x^-3(dx/dt)]/ z (divided both sides by 2z)

When x = 2, z = √(4+1) = √5.

Recall dx/dt = -1 unit/min

dz/dt = [2(-1) - (4/(2)³)(-1)]/√5 = [-2 - 1/2(-1)] / √5 = -3√5/10 (after simplification); this means the distance of the ant from the origin is decreasing when it reaches the point (2,1).