First Principles AI

Solution

Given:

h(t)=-5t^2+20t+30

We want $h(t)\ge 40$ :

-5t^2+20t+30\ge 40

-5t^2+20t-10\ge 0

Divide by $-5$ (reverse inequality):

t^2-4t+2\le 0

Solve $t^2-4t+2=0$ :

\begin{aligned} t&=\frac{4\pm\sqrt{16-8}}{2}\\ &=\frac{4\pm\sqrt{8}}{2}\\ &=\frac{4\pm 2\sqrt{2}}{2}\\ &=2\pm\sqrt{2} \end{aligned}

Since the parabola opens upward, $\le 0$ holds between the roots:

\boxed{2-\sqrt{2}\le t\le 2+\sqrt{2}}

\boxed{0.586\lesssim t\lesssim 3.414\ \text{seconds}}