Two parallel sides of a rectangle are being lengthened at th

Two parallel sides of a rectangle are being lengthened at the rate of 2 in/sec, while the other two sides are shortened in such a way that the figure remains a rectangle with constant area of 50 in2. What is the rate of change of the perimeter when the length of an increasing side is 5 in? Is the perimeter increasing or decreasing?

Solution

l be the increasing length and b be the decreasing length A = lb = 50 bdl/dt - ldb/dt = 0 (negative because b is decreasing) bdl/dt = ldb/dt db/dt = b/l *dl/dt.........(1) perimeter,P = 2l+2b dP/dt = 2dl/dt - 2db/dt dP/dt = 2dl/dt - 2b/l *dl/dt l = 5, dl/dt = 2, b = 10 dP/dt = 4 - 2*10/5 *2 dP/dt = 4-8 dP/dt = -4in/sec.........perimeter is decreasing