The position function s(t)=t3−8t gives the position in miles of a freight train where east is the positive direction and t is measured in ho

Question

The position function s(t)=t3−8t gives the position in miles of a freight train where east is the positive direction and t is measured in hours.

a. Determine the direction the train is traveling when s(t)=0 .
b. Determine the direction the train is traveling when a(t)=0 .
c. Determine the time intervals when the train is slowing down or speed up.

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Rose 3 weeks 2021-11-18T04:38:50+00:00 1 Answer 0 views 0

Answers ( )

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    2021-11-18T04:40:44+00:00

    Answer:

    a. Negative direction when t = 0s and positive direction when t = 2.83s

    b. Negative direction

    c. The train would be slowing down when t is between 0 and 1.633 s then speeding up when t is > 1.633 s

    Step-by-step explanation:

    The velocity function is the derivative of position function

    v(t) = s'(t) = 3t^2 - 8

    The acceleration function si the derivative of velocity function

    a(t) = v'(t) = 6t

    a. When s(t) = 0 then

    t^3 - 8t = 0

    t(t^2 - 8) = 0

    t = 0 or

    t^2 - 8 = 0

    t = \sqrt{8} = 2.83 s

    Plug both of the ts into the velocity function and we have

    v(0) = -8 so negative direction

    v(\sqrt{8}) = 3*8 - 8 = 16 so positive direction

    b. When a(t) = 0 then 6t = 0 so t = 0. v(0) = -8 so negative direction

    c. As a = 6t and t is larger or equal to 0. Then a is also larger or equal to 0. The trains is speeding up if v is positive and slowing down when v is negative

    When v is positive

    v(t) > 0

    3t^2 - 8 > 0

    t^2 > 8/3

    t > \sqrt{8/3} = 1.633 s or t < -\sqrt{8/3} = -1.633 (not possible)

    Similarly, v is negative when t < 1.633 s

    So the train would be slowing down when t is between 0 and 1.633 s then speeding up when t is > 1.633 s

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