For , the curve described by changes over time as its semi-major and semi-minor axes…

MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation

Problem

For

t \ge 0

, the curve described by

\frac{x^2}{a(t)^2} + \frac{y^2}{b(t)^2} = 1

changes over time as its semi-major and semi-minor axes grow according to

a(t) = 1 + 0.5t

and

b(t) = 2 + 0.2t

, where

x

and

y

are measured in millimeters and

t

in seconds. A particle remains on the curve for all

t

. At

t = 4\,\text{s}

, the particle’s coordinates are

(1.5,\,y)

with

y > 0

, and its

x

-coordinate is increasing at a rate of

0.8\,\text{mm/s}

. (a) Obtain an equation that relates

x

y

\frac{dx}{dt}

\frac{dy}{dt}

\frac{da}{dt}

, and

\frac{db}{dt}

for any time

t

. (b) Determine the numerical values of

a

b

\frac{da}{dt}

, and

\frac{db}{dt}

t = 4\,\text{s}

. (c) Using parts (a) and (b), compute

\frac{dy}{dt}

t = 4\,\text{s}

. (d) Let

r(t) = \sqrt{x^2 + y^2}

denote the distance of the particle from the origin. Determine

\frac{dr}{dt}

t = 4\,\text{s}

and state whether the particle is moving toward or away from the origin at that instant.