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 = 8cm and y = 6cm, find the rates of change of (a) the perimeter, and (b) the area of the rectangle.
(a) It is given that the length (x) is decreasing at the rate of 5 cm/minute and the width (y) is increasing at the rate of 4 cm/minute, then we have,
The perimeter (P) of a rectangle is given by:
P = 2( x + y )
= 2(-5 +4) = -2 cm/min
Therefore, the rate of change of the perimeter is -2cm/min.
(b) It is given that the length (x) is decreasing at the rate of 5 cm/minute and the width (y) is increasing at the rate of 4 cm/minute, then we have,
Now, the area (A) of the rectangle is given by
A = x × y
= -5y + 4x
So, when x = 8cm and y = 6cm, then,
Therefore, the rate of change of the area of the rectangle is 2cm2/min.