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/minute. When x = 8 cm and y = 6 cm, find the rates of change of (a) the perimeter, and (b) the area of the rectangle.

Answer 1

It is given that the length of rectangle is x cm which is decreasing at a rate of 5 cm/minute and the width of rectangle is ycm which is increasing at a rate of 4 cm/minute.

(a)

The formula for the perimeter of the rectangle is given by,

P=2( x+y )

Differentiate perimeter P with respect to time t,

dP dt =2 d( x+y ) dt =2( dx dt + dy dt )

As it is given that dx dt =−5 cm/min and dy dt =4 cm/min. Therefore,

dP dt =2( −5+4 ) =−2 cm/min

Thus, the rate of change of perimeter is dP dt =−2 cm/min.

(b)

The formula for the area of rectangle is given by,

A=xy

Differentiate area A with respect to time t,

dA dt =y dx dt +x dy dt

As it is given that dx dt =−5 cm/min and dy dt =4 cm/min. Therefore,

dA dt =( −5y+4x )  cm 2 /min

Substitute x=8cm and y=6cmin the above equation,

dA dt =−5( 6 )+4( 8 ) =2  cm 2 /min

Thus, the area of rectangle is increasing at a rate of 2  cm 2 /min.