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

The rate at which the distance between Charles and Maria is changing can be found using the Pythagorean theorem. The distance between them at any given time is the hypotenuse of a right triangle, with Charles' distance travelled as one side and Maria's distance travelled as the other side. Using this information, we can calculate that the distance is changing at a rate of 28.3 miles every 3 hours.

Step-by-step explanation:

To find the rate at which the distance between Charles and Maria is changing, we can use the Pythagorean theorem. The distance between them at any given time is the hypotenuse of a right triangle, with Charles' distance travelled as the length of one side and Maria's distance travelled as the length of the other side.

Using the formula a^2 + b^2 = c^2, where a and b are the distances travelled by Charles and Maria, respectively, and c is the distance between them, we can substitute in the given information:

a = 5 miles/hour * 3 hours = 15 miles
b = 8 miles/hour * 3 hours = 24 miles
c = √(15^2 + 24^2) = √(225 + 576) = √801 = 28.3 miles

Therefore, the distance between Charles and Maria is changing at a rate of 28.3 miles every 3 hours.

Learn more about Calculating rates of change

Answer 2

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