Mathematics

Problem

Consider a particle moving in the first quadrant along the ellipse

9x^2 + 4y^2 = 36

, where

x

and

y

are measured in centimeters and

t

in seconds. At the instant when the particle has

x

–coordinate

2\,\text{cm}

, its

x

–coordinate is decreasing at a rate of

3\,\text{cm/s}

. a. Part 1: (a) Derive a general relationship among

x

y

\frac{dx}{dt}

, and

\frac{dy}{dt}

that must be satisfied at every instant of the motion. b. Part 2: (b) Using the result from part (a), determine

\frac{dy}{dt}

(in cm/s) when

x = 2

and

y > 0

. c. Part 3: (c) Calculate the instantaneous slope

\frac{dy}{dx}

at that point and interpret its sign in terms of the particle’s vertical direction of motion.