“Two cars start moving from the same point. One travels south at 48 mi/h and the other travels west at 20 mi/h. At what rate is the distance

Question

“Two cars start moving from the same point. One travels south at 48 mi/h and the other travels west at 20 mi/h. At what rate is the distance between the cars increasing four hours later?”

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Samantha 2 weeks 2021-09-12T22:40:15+00:00 1 Answer 0

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    2021-09-12T22:41:40+00:00

    Answer: the rate at which the distance between both cars is increasing is 52 mph

    Step-by-step explanation:

    The direction of movement of both cars forms a right angle triangle. The distance travelled due south and due east by both cars represents the legs of the triangle. Their distance apart after t hours represents the hypotenuse of the right angle triangle.

    Let x represent the length the shorter leg(west) of the right angle triangle.

    Let y represent the length the longer leg(south) of the right angle triangle.

    Let z represent the hypotenuse.

    Applying Pythagoras theorem

    Hypotenuse² = opposite side² + adjacent side²

    Therefore

    z² = x² + y²

    To determine the rate at which the distances are changing, we would differentiate with respect to t. It becomes

    2zdz/dt = 2xdx/dt + 2ydy/dt- – – — – -1

    One travels south at 48 mi/h and the other travels west at 20 mi/h. It means that

    dx/dt = 20

    dy/dt = 48

    Distance = speed × time

    Since t = 4 hours, then

    x = 20 × 4 = 80 miles

    y = 48 × 4 = 192 miles

    z² = 80² + 192² = 6400 + 36864

    z = √43264

    z = 208 miles

    Substituting these values into equation 1, it becomes

    2 × 208 × dz/dt = 2 × 80 × 20 + 2 × 192 × 48

    416dz/dt = 3200 + 18432

    416dz/dt = 21632

    dz/dt = 21632/416

    dz/dt = 52 mph

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