To solve this problem, we will model the path of the soccer ball as a quadratic equation because it follows a parabolic trajectory. Here's the step-by-step solution:

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Step 1: Define the quadratic equation

The general form of the quadratic equation is: $h(t) = at^2 + bt + c$ where:

• $h(t)$ is the height of the ball at time $t$,
• $a, b, c$ are constants to be determined.

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Step 2: Key points and conditions

From the problem, we know:

1. At $t = 0$, $h = 0$ (the ball starts from the ground). $h(0) = 0 \implies c = 0$ So, the equation simplifies to: $h(t) = at^2 + bt$
2. At $t = 1$, $h = 3$: $3 = a(1)^2 + b(1) \implies 3 = a + b$
3. At $t = 4$, $h = 0$ (the ball hits the ground): $0 = a(4)^2 + b(4) \implies 0 = 16a + 4b$ Dividing through by 4: $0 = 4a + b$

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Step 3: Solve for $a$ and $b$

From the two equations:

1. $3 = a + b$,
2. $0 = 4a + b$,

We solve for $a$ and $b$ using substitution:

• From $0 = 4a + b$, rearrange for $b$: $b = -4a$
• Substitute $b = -4a$ into $3 = a + b$: $3 = a - 4a \implies 3 = -3a \implies a = -1$
• Substitute $a = -1$ into $b = -4a$: $b = -4(-1) \implies b = 4$

Thus, the equation for the height of the ball is: $h(t) = -t^2 + 4t$

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Step 4: Find the maximum height

The maximum height occurs at the vertex of the parabola. The time of the vertex is given by: $t = -\frac{b}{2a}$ Here, $a = -1$ and $b = 4$: $t = -\frac{4}{2(-1)} = 2 \, \text{seconds}$

Substitute $t = 2$ into $h(t)$ to find the maximum height: $h(2) = - (2)^2 + 4(2) = -4 + 8 = 4 \, \text{metres}$

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Step 5: Sketch and summarize

The path of the ball is a parabola opening downward. Key points are:

• The ball starts at $(0, 0)$,
• At $t = 1$, $h = 3$,
• The maximum height is $4$ metres at $t = 2$,
• The ball hits the ground at $t = 4$.

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Final Answer:

The maximum height of the soccer ball is: $\boxed{4 \, \text{metres}}$

Would you like me to sketch the parabola or elaborate on any part of the solution?

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Follow-Up Questions:

1. Can you explain how the vertex formula determines the maximum height?
2. Why is the trajectory of the ball modeled as a quadratic equation?
3. What would happen to the equation if the ball was kicked higher initially?
4. How does the time of flight affect the shape of the parabola?
5. How can you verify the solution using physics principles like projectile motion?

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Tip: Always verify the symmetry of the parabola when solving trajectory problems to ensure your equation aligns with given conditions.