Mathematico | Mathematico

At time $t$ seconds, the height of a fighter plane above a radar base is

h = \frac{5}{3} t^{2} - 100 t + 2000

If the plane's height falls below $475 m$ , the plane will be detected by the base.

Determine whether or not the plane is detected, and give the minimum height of the plane below.

Hint

To remove the $\frac{5}{3}$ , you can do

h = \frac{5}{3} [t^{2} - 60 t] + 2000

Now complete the square to find the minimum height of the fighter plane.

Solution

By completing the square,

\begin{aligned} h & = \frac{5}{3} t^{2} - 100 t + 2000 \\ = \frac{5}{3} [t^{2} - 60 t] + 2000 \\ = \frac{5}{3} [(t - 30)^{2} - 900] + 2000 \\ = \frac{5}{3} (t - 30)^{2} - 1500 + 2000 \\ = \frac{5}{3} (t - 30)^{2} + 500 \end{aligned}

At time $t = 30$ , the minmum height of $500 m$ is reached, and the plane does not get detected.