Question

The length x of a rectangle is decreasing at the rate of 2 cm/s and the width y is increasing at the rate of 2 cm/s. When x = 12 cm and y = 5 cm, find the rate 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
$therefore space space space space space space dx over dt space equals space minus 2 space cm divided by straight s space space space and space dy over dt space equals space 2 space cm divided by straight s$ ...(1)
(a) The perimeter P of the rectangle is given by
$straight P space equals space 2 left parenthesis straight x plus straight y right parenthesis$
$therefore space space space space space space dP over dt space equals space 2 open parentheses dx over dt plus dy over dt close parentheses space equals space 2 space left parenthesis negative 2 plus 2 right parenthesis space space space space space space space space space space space space space space space space space space space space space space space open square brackets because space space space of space left parenthesis 1 right parenthesis close square brackets therefore space space space space space space space dP over dt space equals space 0 space cm divided by sec$
(b) The area A of the rectangle is given by
$straight A space equals space xy$
$therefore space space space space space dA over dt space equals space straight x dy over dt plus straight y dx over dt space equals space 12 left parenthesis 2 right parenthesis space plus space 5 left parenthesis negative 2 right parenthesis space space space space space space space space space space space space space space space open square brackets therefore space space straight x space equals space 12 comma space space space straight y space equals space 5 space and space of space left parenthesis 1 right parenthesis close square brackets space space space space space space space space space space space space space space space space space space equals 24 minus 10 space equals space 14 space cm squared divided by straight s$

For Daily Free Study Material Join wiredfaculty

Application Of Derivatives

The length x of a rectangle is decreasing at the rate of 2 cm/s and the width y is increasing at the rate of 2 cm/s. When x = 12 cm and y = 5 cm, find the rate of change of (a) the perimeter and (b) the area of the rectangle.

Some More Questions From Application of Derivatives Chapter

The radius of a circle is increasing uniformly at the rate of 4 cm per second Find the rate at which the area of the circle is increasing when the radius is 8 cm.

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

Phone Number

Email Address

Location