### Home > APCALC > Chapter 8 > Lesson 8.1.4 > Problem8-44

8-44.

The Pi hotel is in the town of Accelerton. The hotel features a grand fireworks display every 4th of July. After launching a firework from the top of the building, the projectile reaches its maximum height where it explodes, amazing the crowds on the ground. The function that models the height of the shell for the firework (in feet) at any time $t$ (in seconds) is given by the function $s(t) = -16t^2 + 120t + 136$. Homework Help ✎

1. How tall is the hotel?

Since the distance function tracks the height of the shell, and the shell was launched from the top of the hotel, where was the shell at $t = 0$?

2. What is the initial velocity of the projectile?

You can take the derivative and evaluate at $t = 0$.
Or you can analyze the position function:

$s(t)=\frac{1}{2}at^{2}+v_{0}t+s_{0}$

3. In order to determine the length of the fuse, the organizers need to know when the projectile will be at its maximum height. At what time should the firework explode?

Notice that the distance function is a parabola. Use algebra or calculus to determine the $x$-value of the vertex of this parabola.

4. How high will the explosion occur?

Recall that a local maximum (or local minimum) is a $y$-value, not an $x$-value. After all, max and min mean highest and lowest, which are vertical measurements ($y$-axis measurements).

5. Oh no! The timing device failed! The shell is falling towards the ground. If the shell is traveling faster than $150$ ft/sec it will explode on contact with the ground. Will the shell explode?

Given that $s(0) ≠ 0$, is it reasonable to assume that the ground is located at$s(t) = 0$?