Question

A bug moves along the curve y=4− 16x^2. Distance is measured in feet. The bug's y-coordinate is decreasing at 20 ft/sec when it reaches the point (4,3). How fast is its x-coordinate changing?

a) −5ft/sec
b) −4ft/sec
c) −3ft/sec
d) −2ft/sec

Answer 1

The bug's x-coordinate is changing at a rate of -1/128 ft/sec.

Step-by-step explanation:

To find how fast the bug's x-coordinate is changing, we need to use related rates. We know that the bug's y-coordinate is decreasing at 20 ft/sec when it reaches the point (4,3). We can differentiate the equation of the curve to get dy/dx=-32x. Plugging in x=4 gives us the rate of change of y with respect to x at that point. dy/dx=-32(4)=-128 ft/sec.

Since the bug is moving along the curve and not changing its y-coordinate, we know that the bug's y-coordinate is changing at the same rate as the curve. Therefore, the rate of change of the x-coordinate is the negative reciprocal of the rate of change of the y-coordinate. So the rate of change of the x-coordinate is 1/(-128)=-1/128 ft/sec.