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

Explanation

`(dx)/(dt) = - 5 cm//min, (dy)/(dt) = 4 ` cm/min
(a) Let the perimeter of rectangle be P.
`:. P = 2(x+y)`
` rArr (dP)/(dt) = 2 ((dt)/(dt)+(dy)/(dt)) = 2 (-5 + 4) =-2`
`:.` The perimeter of rectangle is decreasing at the rate of 2 cm/min.
(b) Let area of rectangle be A.
`:. A xy`
`rArr (dA)/(dt) = d/(dt) (xy) = x* (dy)/(dt) + y (dx)/(dt)`
at x = 8 and y = 6
`(dA)/(dt) = 8 xx 4+ 6 xx (-5) = 2`
` :. ` Area of rectangle is increasing at the rate of `2 cm^(2)//min`.