Mathematics

Problem

A rectangle has length

x(t)

cm and width

y(t)

cm, both varying with time

t

. Answer the following questions neatly showing all steps. a. (i) Write the formula for the perimeter

P(t)

of the rectangle in terms of

x(t)

and

y(t)

. b. (ii) Differentiate your expression in part (i) with respect to

t

and obtain a relation between

dP/dt

dx/dt

and

dy/dt

. c. (iii) At a certain instant the length is increasing at

2\,\text{cm s}^{-1}

while the width is decreasing at

2\,\text{cm s}^{-1}

. State, with reason, whether the perimeter is increasing, decreasing or remaining constant at that instant. d. (iv) Derive the general condition connecting

dx/dt

and

dy/dt

for which the perimeter of the rectangle does not change with time.

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