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 2 cm/minute. When x = 10 cm and y = 6 cm, find the rate of change of (a) the perimeter and (b) the area of rectangle

Solution

Since the length x is decreasing and the width y is increasing
$therefore space space space space space dx over dt space equals space minus 3 space space cm divided by minute space space space and space space dy over dt space equals space 2 cm divided by minute$ ...(1)
(a) The perimeter P of the rectangle is given by
$straight P space equals space 2 left parenthesis straight x plus straight y right parenthesis$
$therefore space space space space space space dP over dt space equals space 2 space open parentheses dx over dt plus dy over dt close parentheses space equals space 2 left parenthesis negative 3 plus 2 right parenthesis space equals space minus 2 space cm divided by minute$
(b) The area A of the rectangle is given by
A = xy
$therefore space space space space space space dx over dt space equals space straight x dy over dt plus straight y dx over dt space equals space 10 left parenthesis 2 right parenthesis space plus space 6 left parenthesis negative 3 right parenthesis space space space space space space space space space space space open square brackets because space space space straight x space equals space 10 comma space space straight y space equals space 6 space and space because space space of space left parenthesis 1 right parenthesis close square brackets$
$equals 20 minus 18 space equals space 2 space cm squared divided by minute$

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Application Of Derivatives

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 2 cm/minute. When x = 10 cm and y = 6 cm, find the rate of change of (a) the perimeter and (b) the area of 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.

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