To determine the maximum height of the cannonball, we need to analyze the quadratic function given in the graph, which is:

$h(t) = -16t^2 + 144t$

This is a typical projectile motion equation where:

• The coefficient of $t^2$, $-16$, represents the effect of gravity (in feet per second squared),
• The linear term $144t$ represents the initial velocity (144 feet per second).

The problem states that the cannonball reaches its maximum height at $t = 4.5$ seconds. To find the maximum height, we substitute $t = 4.5$ into the equation $h(t)$.

$h(4.5) = -16(4.5)^2 + 144(4.5)$

First, calculate the squared term:

$4.5^2 = 20.25$

Now substitute and simplify:

$h(4.5) = -16(20.25) + 144(4.5)$ $h(4.5) = -324 + 648$ $h(4.5) = 324 \, \text{feet}$

Thus, the maximum height the cannonball reached is 324 feet.

Do you want more details or have any questions? Here are five related questions you might want to explore:

1. How is the initial velocity related to the height in projectile motion?
2. What happens if we change the initial velocity in the equation?
3. How do you determine the time at which the cannonball reaches the ground?
4. How does the acceleration due to gravity affect the trajectory?
5. Can we derive the maximum height using the vertex form of a quadratic equation?

Tip: In projectile motion problems, the maximum height is reached when the derivative of the height function equals zero, indicating the vertex of the parabola.