Related rates: the hypotenuse of an isosceles right triangle is increasing at a rate of 2mm/s. at what rate is the area of the triangle increasing when the length of one of the equal sides is 10mm? related rates: the hypotenuse of an isosceles right triangle is increasing at a rate of 2mm/s. at what rate is the area of the triangle increasing when the length of one of the equal sides is 10mm?

The answer is It is increasing at 2√2 mm/s.

Below is the solution:

Call the length of the legs x and the hypotenuse z, then we have z2=2x2(Pythagorean Theorem)

We know that the rate of change of x with respect to time, t, is dx/dt=2 mm/s.

We want to find dz/dt at the instant when x=10 mm

Differentiate z2=2x2 implicitly with respect to time, t:

d/dt(z2) = d/dt(2x2)
2z dz/dt = 4x dx/dt
z dz/dt = 2x dx/dt

z=10√2 mm
10√2 mm)dz/dt = 2(10 mm)(2 mm/s)

dz/dt = 2√2 mm/s

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