Example 4 The length x of a rectangle is decreasing at the rate of 3 cm/minute and
the width y is increasing at the rate of 2cm/minute When x =10cm and y = 6cm, find
the rates of change of (a) the perimeter and (b) the area of the rectangle Solution Since the length x is decreasing and the width y is increasing with respect to
time, we have
3 cm/min
dx
dt = −
and
2 cm/min
dy
dt =
(a)
The perimeter P of a rectangle is given by
P = 2 (x + y)
Therefore
dP
dt  = 2
2
3
2
2
dx
dt
dy
+dt


 =
− +
= −
(
)
cm/min
(b)
The area A of the rectangle is given by
A = x y
Therefore
dA
dt  = dx
dy
y
x
dt
dt
⋅
+
⋅
= – 3(6) + 10(2)
(as x = 10 cm and y = 6 cm)
= 2 cm2/min
Rationalised 2023-24
 MATHEMATICS
150
Example 5 The total cost C(x) in Rupees, associated with the production of x units of
an item is given by
C(x) = 0