Question

the lengths of two legs of a right triangle depend on time. One, whose length is x increase at the rate of 5 feet per second, while the other, of length y, decreases at the rate of 6 feet per second. At what rate is the hypotenuse changing when x=3 and y=4? is the hypotenuse increasing or decreasing then?

Answer 1

given: dx/dt = 5 , dy/dt = -6
when x=3 and y = 4 , hypotenuse = 5 , (the infamous 3-4-5 right-angled triangle)

H^2 = x^2 + y^2
2H dH/dt = 2x dx/dt + 2y dy/dt
dH/dt = (x dx/dt + y dy/dt)/H
= (3(5) + 4(-6))/5
= -9/5

at that moment of time, the hypotenuse is <b>decreasing</b> at a rate of 9/5 ft/s