#### Example 6.89.

The Chamber of Commerce in River City plans to put on a Fourth of July fireworks display. City regulations require that fireworks at public gatherings explode higher than \(800\) feet above the ground. The mayor particularly wants to include the Freedom Starburst model, which is launched from the ground. Its height after t seconds is given by

\begin{equation*}
h = f(t) = 256t - 16t^2
\end{equation*}

When should the Starburst explode in order to satisfy the safety regulation?

## Solution.

We can get an approximate answer to this question by looking at the graph of the rocket’s height, shown below.

When is the rocket’s height greater than 800 feet, or, in mathematical terms, for what values of \(t\) is \(h \gt 800\text{?}\) The answer to this question is the solution of the inequality

\begin{equation*}
256t - 16t^2 \gt 800
\end{equation*}

Points on the graph with \(h \gt 800\) are shown in color, and the \(t\)-coordinates of those points are marked on the horizontal axis. If the Freedom Starburst explodes at any of these times, it will satisfy the safety regulation.

From the graph, the safe time interval runs from approximately \(4.25\) seconds to \(11.75\) seconds after launch. The solution of the inequality is the set of all \(t\)-values greater than \(4.25\) but less than \(11.75\text{.}\)