Mathematics

Problem

A particle moves in the plane so that its coordinates

(x, y)

, measured in centimeters, satisfy the equation

(x^2 + y^2)^2 = 16(x^2 - y^2)

for all time

t

. At the instant when the particle is at the point

(3, 1)

, its

y

-coordinate is increasing at a rate of

2\,\text{cm/s}

. a. Part (a): Establish an equation relating

\frac{dx}{dt}

and

\frac{dy}{dt}

that holds while the particle moves along the given curve. b. Part (b): Compute

\frac{dx}{dt}

at the point

(3, 1)

. c. Part (c): The particle’s speed is

v = \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2}

. Determine

v

at the given instant. d. Part (d): Find the angle

\theta

(in radians) between the velocity vector and the positive

x

-axis at that instant, where

0 < \theta < \frac{\pi}{2}

, and justify your answer.