## Maximum Distance of Particle From Origin

Suppose the position of a particle on the
$x$
axis is
$x=80t-5t^2, \: x \gt 0$
m.
What will be the maximum distance of the particle from the origin?
For this particular example we can write
\begin{aligned} x &= 80t-5t^2=-5(t^2-16t) \\ &= -5((t-8)^2-(-8)^2) \\ &= -5((t-8)^2-64) \\ &= 320-5(t-8)^2 \ge 320 \end{aligned}

The maximum distance is 320m.