A rectangular metal sheet has its length and width changing with time

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

Problem

A rectangular metal sheet has its length

x\,(\mathrm{cm})

and width

y\,(\mathrm{cm})

changing with time. At a certain instant the length is increasing at

1.5\,\mathrm{cm\,s^{-1}}

while the width is decreasing at

0.5\,\mathrm{cm\,s^{-1}}

. When

x=20\,\mathrm{cm}

and

y=10\,\mathrm{cm}

, find the instantaneous rate of change of the perimeter of the sheet. (i) Write the expression for the perimeter

P

of the rectangle in terms of

x

and

y

. (ii) Differentiate

P

with respect to time

t

to obtain

\dfrac{dP}{dt}

in terms of

x,\,y,\,\dfrac{dx}{dt}

and

\dfrac{dy}{dt}

. (iii) Substitute the given numerical values to calculate

\dfrac{dP}{dt}

, mentioning the correct unit.