Application of Derivatives

Question

The length x of a rectangle is decreasing at the rate of 5 cm/minute and the width y is increasing at the rate of 4 cm/minutc. When x = 8 cm and y = 6 cm, 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
$therefore space space space dx over dt space equals space minus 5 space cm divided by minute space space space and space dy over dt space equals space 4 space cm divided by minute$ ...(1)
(a) The perimeter P of the rectangle is given by
$straight P space equals space 2 left parenthesis straight x plus straight y right parenthesis$
$therefore space space space space space space space space dP over dt space equals space 2 space open parentheses dx over dt plus dy over dt close parentheses space equals space 2 left parenthesis negative 5 plus 4 right parenthesis space space space space space space space space space space space space space space space space space space space space space space space equals space minus 2 space cm divided by minute space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space open square brackets because space space space space of space left parenthesis 1 right parenthesis close square brackets$
(b) The area A of the rectangle is given by
$straight A space equals space xy$
$therefore space space space space space dA over dt space equals space straight x dy over dt plus straight y dx over dt$
$equals space 8 left parenthesis 4 right parenthesis space plus space 6 left parenthesis negative 5 right parenthesis space space space space space space space space space space space space space space space space space space space space space open square brackets because space space space straight x space equals space 8 comma space space space straight y space equals space 6 space and space space because space space of space left parenthesis 1 right parenthesis close square brackets space equals space 32 minus 30 space equals space 2 space cm squared divided by minute.$

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Application Of Derivatives

The length x of a rectangle is decreasing at the rate of 5 cm/minute and the width y is increasing at the rate of 4 cm/minutc. When x = 8 cm and y = 6 cm, find the rates of change of (a) the perimeter, and (b) the area of the rectangle.

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