Two hikers, Charles and Maria, begin at the same location and travel in perpendicular directions. Charles travels due north at a rate of 5 miles per hour. Maria travels due west at a rate of 8 miles per hour. At what rate is the distance between Charles and Maria changing exactly 3 hours into the hike?

As Charles and Maria travel in perpendicular directions, these directions could be considered as two legs ( x and y ) of a right triangle and distance (L) between them as the hypotenuse, therefore, according to Pythagoras theorem

L² = x² + y²

And as all ( L , x , and y ) are function of time, we apply differentiation in both sides of the equation to get

2* L d(L)/dt = 2*x*d(x)/dt + 2*y*d(y)/dt (1)

In equation (1) we know:

d(x)/dt = 8 miles/per hour ( Maria )

d(y)/dt = 5 miles /per hour (Charles)

In 3 hours time Maria has travel 3*8 = 24 miles

And Charles 5*3 = 15 miles

Then at that time L is equal to

L = √ 24² + 15² ⇒ L = √ 576 + 225 ⇒ L = √801 ⇒ L =28,30 miles

Then plugging these values in equation (1)

2* L d(L)/dt = 2*x*d(x)/dt + 2*y*d(y)/dt

2* 28.30 * d(L)/dt = 2*24*8 + 2* 15*5

56.6 *d(L)/dt = 384 + 150

d(L)/dt = 534/56,6

d(L)/dt = 9,43 miles per hour