Two cars start moving from the same point. One travels south at 60 miyh and the other travels west at 25 miyh. At what rate is the distance

Question

Two cars start moving from the same point. One travels south at 60 miyh and the other travels west at 25 miyh. At what rate is the distance between the cars increasing two hours later?

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Thiên Hương 3 years 2021-08-27T00:44:29+00:00 1 Answers 22 views 0

Answers ( )

  1. Jezebel
    0
    2021-08-27T00:45:42+00:00
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    Answer:

    65 m/h

    Explanation:

    Let the distance of the car moving south be y.

    Let the distance of the car moving west be x.

    Let the distance between the two cars be a.

    These three distances can be represented as a right angled triangle. So we can say:

    a^2 = x^2 + y ^2

    Let us differentiate with respect to time, since the distances are changing with respect to time:

    2a\frac{da}{dt} = 2x\frac{dx}{dt} + 2y\frac{dy}{dt} \\\\=>a\frac{da}{dt} = x\frac{dx}{dt} + y\frac{dy}{dt}__________(1)

    da/dt = rate of change of distance between two cars

    The speed of the car moving south (dy/dt) is 60 m/h and the speed of the car moving west (dx/dt) is 25 m/h.

    Therefore:

    dy/dt = 60 m/h and dx/dt  = 25 m/h

    After two hours, the distance of the two cars will be:

    y = 2 * 60 = 120 miles

    x = 2 * 25 = 50 miles

    Therefore:

    a^2 = 50^2 + 120^2\\\\a^2 = 2500 + 14400 = 16900\\\\a = \sqrt{16900}\\ \\a = 130 miles

    From (1):

    130(da/dt) = 50(25) + 120(60)

    130(da/dt) = 1250 + 7200 = 8450

    da/dt = 8450/130 = 65 m/h

    Therefore, after two hours, the distance between the two cars is changing at a rate of 65 m/h.

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