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?

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

6,250

step-by-step explanation:

to get the area, multiply 360 by 160.(57,600)

to find out how many bags of fertilizer it will take, first figure out how many square feet one bag can hold. divide 1000 by 8(125) to get the unit rate(how many square feet one pound of fertilizer can hold)and multiply it by 50(6250). if i haven't made a mistake, that is the answer.

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