The rate of change of the hypotenuse is ______ meters per minute.

Question

The base of a right triangle is increasing at a rate of 4 meters per minute and the height is increasing at a rate of 3 meters per minute.

When the base is 10 meters and the height is 16 meters, then how fast is the hypotenuse changing?

The rate of change of the hypotenuse is ______ meters per minute.

(Enter your answer as a integer or as a decimal number rounded to 2 places.)

Answer 1

Let x = base, y = height, and z = hypotenuse

z² = x² + y² When x = 10 and y = 16, z = √(100 + 256) = √356 = 2√89

2z(dz/dt) = 2x(dx/dt) + 2y(dy/dt)

dz/dt = [x(dx/dt) + y(dy/dt)] / z = [40 + 48] / (2√89) = 44 / √89