Question

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?

Answer 1

Answer:

d(L)/dt   = 9,43 miles per hour

Step-by-step explanation:

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