The position of a particle moving along the x-axis is given by $x (t) = t^{3} - 9 t^{2} + 24 t$ $x(t)=t^3-9t^2+24t$ meters where t is in seconds, $t \geq 0$ $t>=0$ a) Draw a sign diagram for the particle's velocity and acceleration functions.?

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The position of a particle moving along the x-axis is given by $x (t) = t^{3} - 9 t^{2} + 24 t$ $x(t)=t^3-9t^2+24t$ meters where t is in seconds, $t \geq 0$ $t>=0$ a) Draw a sign diagram for the particle's velocity and acceleration functions.?

$t \geq 0$ $t>=0$

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Answer 1 · 2018-01-21T10:33:28

See the explanation below

Explanation:

The velocity is the derivative of the position

$x (t) = t^{3} - 9 t^{2} + 24 t$ $x(t)=t^3-9t^2+24t$

$v (t) = 3 t^{2} - 18 t + 24 = 3 (t^{2} - 6 t + 8)$ $v(t)=3t^2-18t+24=3(t^2-6t+8)$

$= 3 (t - 2) (t - 4)$ $=3(t-2)(t-4)$

The acceleration is the derivative of the velocity

$v (t) = 3 t^{2} - 18 t + 24$ $v(t)=3t^2-18t+24$

$a (t) = 6 t - 18 = 6 (t - 3)$ $a(t)=6t-18=6(t-3)$

The sign chart for the velocity is as follows :

$a a a a$ $color(white)(aaaa)$ $t$ $t$ $a a a a a$ $color(white)(aaaaa)$ $0$ $0$ $a a a a a a a a$ $color(white)(aaaaaaaa)$ $2$ $2$ $a a a a a a a a$ $color(white)(aaaaaaaa)$ $4$ $4$ $a a a a a a$ $color(white)(aaaaaa)$ $+ \infty$ $+oo$

$a a a a$ $color(white)(aaaa)$ $t - 2$ $t-2$ $a a a a a a$ $color(white)(aaaaaa)$ $-$ $-$ $a a$ $color(white)(aa)$ $0$ $0$ $a a a$ $color(white)(aaa)$ $+$ $+$ $a a a a a a$ $color(white)(aaaaaa)$ $+$ $+$

$a a a a$ $color(white)(aaaa)$ $t - 4$ $t-4$ $a a a a a a$ $color(white)(aaaaaa)$ $-$ $-$ $a a$ $color(white)(aa)$ #color(white)(aaaa)-# $a a a$ $color(white)(aaa)$ $0$ $0$ $a a$ $color(white)(aa)$ $+$ $+$

$a a a a$ $color(white)(aaaa)$ $v (t)$ $v(t)$ $a a a a a a$ $color(white)(aaaaaa)$ $+$ $+$ $a a a$ $color(white)(aaa)$ $0$ $0$ $a a a$ $color(white)(aaa)$ $-$ $-$ $a a a$ $color(white)(aaa)$ $0$ $0$ $a a$ $color(white)(aa)$ $+$ $+$

$a a a a$ $color(white)(aaaa)$ $s (t)$ $s(t)$ $a a a a a a$ $color(white)(aaaaaa)$ $↗$ $↗$ $a a$ $color(white)(aa)$ $20$ $20$ $a a a$ $color(white)(aaa)$ $↘$ $↘$ $a a$ $color(white)(aa)$ $16$ $16$ $a a$ $color(white)(aa)$ $↗$ $↗$

The sign chart fot the acceleration is as follows :

$a a a a$ $color(white)(aaaa)$ $Interval$ $"Interval"$ $a a a a$ $color(white)(aaaa)$ $(0, 3)$ $(0,3)$ $a a a a$ $color(white)(aaaa)$ $(3, + \infty)$ $(3,+oo)$

$a a a a$ $color(white)(aaaa)$ $a (t)$ $a(t)$ $a a a a a a a a a$ $color(white)(aaaaaaaaa)$ $-$ $-$ $a a a a a a a$ $color(white)(aaaaaaa)$ $+$ $+$

$a a a a$ $color(white)(aaaa)$ $s (t)$ $s(t)$ $a a a a a a a a a$ $color(white)(aaaaaaaaa)$ $\cap$ $nn$ $a a a a a a a$ $color(white)(aaaaaaa)$ $\cup$ $uu$

graph{(y-(x^3-9x^2+24x))(y-(3x^2-18x+24))(y-(6x-18))=0 [-23.9, 41.07, -6.7, 25.78]}

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The position of a particle moving along the x-axis is given by $x (t) = t^{3} - 9 t^{2} + 24 t$ $x(t)=t^3-9t^2+24t$ meters where t is in seconds, $t \geq 0$ $t>=0$ a) Draw a sign diagram for the particle's velocity and acceleration functions.?

$t \geq 0$ $t>=0$

1 Answer

Explanation:

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Humanities

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The position of a particle moving along the x-axis is given by x(t)=t^3-9t^2+24tx(t)=t3−9t2+24tx(t)=t^3-9t^2+24t meters where t is in seconds, t>=0t≥0t>=0 a) Draw a sign diagram for the particle's velocity and acceleration functions.?

t>=0t≥0t>=0

1 Answer

Explanation:

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Impact of this question

The position of a particle moving along the x-axis is given by $x (t) = t^{3} - 9 t^{2} + 24 t$ $x(t)=t^3-9t^2+24t$ meters where t is in seconds, $t \geq 0$ $t>=0$ a) Draw a sign diagram for the particle's velocity and acceleration functions.?

$t \geq 0$ $t>=0$