A particle travels in the first quadrant along the curve defined implicitly by

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

Problem

xy + y^3 = 10

, where

x

and

y

are measured in inches and

t

in seconds. At the instant when the particle’s coordinates are

(2, 2)

, its x-coordinate is decreasing at a rate of

0.6\,\text{in/s}

. (a) Formulate an equation that connects

dx/dt

and

dy/dt

at any point on the curve. (b) Determine

dy/dt

(in

\text{in/s}

) at the given point. (c) Compute the instantaneous rate of change of the product

xy

with respect to time at that instant. (d) The particle continues along the curve until its y-coordinate reaches

2.5\,\text{in}

. Estimate the time required for this change by assuming

dx/dt

and

dy/dt

remain approximately constant over the interval, and briefly discuss the reasonableness of this assumption.