Apply the seven-step formula-based strategy to determine when a projectile reaches a given height, using the projectile motion formula and the Quadratic Formula. Unlike geometry applications where one solution is discarded, this problem produces two valid solutions because the object passes through the target height twice — once on the way up and once on the way down.

Problem: A firework is shot upward with an initial velocity of 130 feet per second. How many seconds will it take to reach a height of 260 feet? Round to the nearest tenth of a second.

1. Read the problem.
2. Identify: The number of seconds (time) for the firework to reach 260 feet.
3. Name: Let $t$ = the number of seconds.
4. Translate: Write the projectile motion formula and substitute the known values. The initial velocity is $v_0 = 130$ ft/sec and the target height is $h = 260$ ft:

$h = -16t^2 + v_0 t$

$260 = -16t^2 + 130t$

5. Solve: This is a quadratic equation. Rewrite it in standard form by moving all terms to one side:

$16t^2 - 130t + 260 = 0$

Identify the coefficients: $a = 16$, $b = -130$, $c = 260$. Substitute into the Quadratic Formula:

$t = \frac{-(-130) \pm \sqrt{(-130)^2 - 4(16)(260)}}{2(16)}$

Simplify inside the square root: $(-130)^2 = 16{,}900$ and $4(16)(260) = 16{,}640$, so $16{,}900 - 16{,}640 = 260$:

$t = \frac{130 \pm \sqrt{260}}{32}$

Since $\sqrt{260}$ does not simplify to a whole number, write the two solutions and approximate:

$t = \frac{130 + \sqrt{260}}{32} \approx 4.6 \quad \text{or} \quad t = \frac{130 - \sqrt{260}}{32} \approx 3.6$

6. Check: Left to the student.
7. Answer: The firework will go up and then fall back down. As the firework rises, it reaches 260 feet after approximately 3.6 seconds. It also passes that height on the way down at approximately 4.6 seconds.

This example differs from previous quadratic applications in an important way: both solutions are physically meaningful. In geometry problems (such as finding a length or width), a negative solution is discarded because measurements cannot be negative. Here, however, both positive time values correspond to real moments in the firework's flight — the smaller value is when the firework is ascending through 260 feet, and the larger value is when it is descending past that same height after reaching its peak.