Question

Given the position of the particle and using interval notation, when is the particle moving left and when is it moving right

Answer 1

we have the expression

$\frac{2}{3}t^3-\frac{9}{2}t^2-18t$

Factor (2/3)t

$\frac{2}{3}t(t^2-\frac{27}{4}t^{}-27)$

Solve the quadratic equation

using the formula

a=1

b=-27/4

c=-27

$t=\frac{-(-\frac{27}{4})\pm\sqrt[]{(-\frac{27}{4})^2-4(1)(-27)}}{2(1)}$$t=\frac{\frac{27}{4}\pm\sqrt[]{\frac{2457}{16}}}{2}$$t=\frac{27}{8}\pm\frac{\sqrt[]{2457}}{8}$

The values of t are

t=-2.82 and t=9.57

therefore

the intervals are

(-infinite, -2.82)