Answer
$c=9.0m/s^2$
Work Step by Step
Remember that $$F(t)=ma(t)$$ according to Newton's second law. Therefore, if $a(t)=\frac{d^2}{dt^2}x(t)$, acceleration can be found by using the power rule to differentiate. This yields $$x'(t)=4.0m/s + 2ct-(6.0m/s^3)t^2$$ $$x''(t)=a(t)=2c-(12m/s^3)t$$ Solving for acceleration when $t=3.0s$ yields an acceleration of $$a(3.0)=2c-(12m/s^3)(3.0s)=2c-36m/s^2$$ Solving Newton's second law for $a(t)$ yields $$a(t)=\frac{F(t)}{m}$$ Substituting known values of $m=2.0kg$ and $F(3.0)=-36N$ (negative direction of the axis) yields $$a(3.0)=\frac{-36N}{2.0kg}=-18m/s^2$$ Substituting the acceleration equation for a(t) at t=3.0s yields $$-18m/s^2=2c-(12m/s^3)(3.0s)=2c-36m/s^2$$ Solving for $c$ yields $$2c=18m/s^2$$ $$c=9.0m/s^2$$
