contestada

The position function s(t)=t3−8ts(t)=t3−8t gives the position in miles of a freight train where east is the positive direction and tt is measured in hours. determine the direction the train is traveling when s(t)=0.s(t)=0. determine the direction the train is traveling when a(t)=0.a(t)=0. determine the time intervals when the train is slowing down or speeding up.

Respuesta :

we are given

[tex] s(t)=t^3-8t [/tex]

(a)

we can set s(t)=0

and then we can solve for t

[tex] t^3-8t=0 [/tex]

[tex] t(t^2-8)=0 [/tex]

[tex] t(t-\sqrt{8}) (t+\sqrt{8})=0 [/tex]

so, we get

[tex] t=0,t=\sqrt{8} [/tex]............Answer

(b)

Firstly, we will find a(t)

we know that a(t) is second derivative of s(t)

[tex] s'(t)=3t^2-8*1 [/tex]

now, we can find derivative again

[tex] s''(t)=3*2t^1-8*0 [/tex]

[tex] a(t)=6t [/tex]

now, we can set it to 0

and then we can solve for t

[tex] a(t)=6t=0 [/tex]

[tex] t=0 [/tex]............Answer


(c)

Firstly, we will set s'(t)=0 and a(t)=0

and then we solve for t

[tex] s'(t)=3t^2-8*1=0 [/tex]

[tex] t=1.63299 [/tex]

[tex] t=0 [/tex]

now, we can draw number line

and then we can check sign of s'(t) and s''(t) on each intervals

same sign means speeding up

different sign means speeding down

speeding up interval:

[tex] (1.63299,\infty) [/tex]

speeding down interval:

[tex] (0, 1.63299) [/tex]

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