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16 April, 02:59

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?

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  1. 16 April, 04:38
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    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
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