Let's solve the problem step by step using the given information from the uploaded image.

Observations from the Graph

The equation of the parabola can be written as: $H(t) = a(t - t_1)(t - t_2)$ Where:

• $t_1 = 0$ (the first root),
• $t_2 = 5$ (the second root),
• Vertex is at $(2.5, 20)$.

The vertex form of a parabola is: $H(t) = a(t - h)^2 + k$ Where:

• $h = 2.5$ (the time of the vertex),
• $k = 20$ (the maximum height of the vertex).

Using the vertex form, expand to find the standard equation of the parabola.

Step 1: Find the coefficient $a$

We know that the height $H(t) = 0$ at $t = 0$ and $t = 5$. Substituting one of these points into the vertex form:

$0 = a(0 - 2.5)^2 + 20$

$0 = 6.25a + 20$

$a = -\frac{20}{6.25} = -3.2$

So the equation becomes: $H(t) = -3.2(t - 2.5)^2 + 20$

Part (a): If the time is $t = 4.5$, find the height.

Substitute $t = 4.5$ into the equation: $H(4.5) = -3.2(4.5 - 2.5)^2 + 20$ $H(4.5) = -3.2(2)^2 + 20$ $H(4.5) = -3.2(4) + 20$ $H(4.5) = -12.8 + 20 = 7.2$

The height of the firework at $t = 4.5$ seconds is $7.2 \, \text{m}$.

Part (b): If the height is $10$, find the time.

We solve: $10 = -3.2(t - 2.5)^2 + 20$ Rearrange: $-3.2(t - 2.5)^2 = 10 - 20$ $-3.2(t - 2.5)^2 = -10$ $(t - 2.5)^2 = \frac{-10}{-3.2} = 3.125$ Take the square root of both sides: $t - 2.5 = \pm \sqrt{3.125}$ $t - 2.5 = \pm 1.77$ Solve for $t$: $t = 2.5 + 1.77 = 4.27 \, \text{s} \quad \text{or} \quad t = 2.5 - 1.77 = 0.73 \, \text{s}.$

The firework reaches a height of $10 \, \text{m}$ at $t = 0.73 \, \text{s}$ and $t = 4.27 \, \text{s}$.

Final Answers:

(a) The height at $t = 4.5 \, \text{s}$ is 7.2 m.
(b) The times when the height is $10 \, \text{m}$ are $0.73 \, \text{s}$ and $4.27 \, \text{s}$.

Would you like further clarification on any step?

Follow-up Questions:

1. How can we derive the vertex form of a parabola from its roots and vertex?
2. Why is the coefficient $a$ negative in this equation?
3. How can we confirm the symmetry of the parabola using its equation?
4. What are other methods to solve quadratic equations besides factoring and completing the square?
5. How does the discriminant help in understanding the solutions to a quadratic equation?

Tip:

Always verify your solutions graphically if possible to confirm the correctness of the computed roots or heights.