Conic Sections: Parabola - Purpose Of This Discussion

Conic Section1parabolathe Purpose Of This Discussion Is To Find An Eq

Conic Section 1 Parabola The purpose of this discussion is to: find an equation to a real life situation such as the problem described below I want to see your ideas, and perspectives. Suppose the staging platform for a fireworks display is 6 ft above ground, and the mortars leave the platform at 96 ft/sec. The height of mortars h(t) (in feet) can be modeled by h(t)=-16t^2 + 96t + 6, where t is the time in seconds after launch: Find the following 1. If the fuses are set for 3 sec after launch, at what height will the fireworks explode? 2. Will the fireworks explode at their maximum height? Explain Instruction: 1. Discuss the fact why it is a parabola problem. 2. Set up an equation of the form: h(t) = a (t - h)^2 + k 3. Use the vertex formula to write the equation and solve the problem above. 4. Respond to at least two students post. 5. Discuss their solutions, why you agree or disagree with them.

Paper For Above instruction

The problem presented involves determining the characteristics of a parabola that models the height of fireworks over time, illustrating a classic quadratic scenario rooted in physics and mathematics. This scenario uses a quadratic function, which is inherently a parabola, because the path of the fireworks follows a projectile motion influenced by gravity, with an initial velocity and height, resulting in a symmetric, curved trajectory typical of quadratic functions.

Firstly, it is crucial to understand why this is a parabola problem. Parabolas are the graphs of quadratic functions, which can be expressed in the form y = ax^2 + bx + c. In the context of projectile motion, such as fireworks, the height of the object over time follows a quadratic function because of the acceleration due to gravity acting downward, causing the initial upward velocity to be gradually offset, creating a curved trajectory. The general form of the height function, h(t) = -16t^2 + 96t + 6, embodies this quadratic nature, with the coefficient of t^2 being negative, indicating a downward opening parabola, consistent with what occurs in real projectile motion.

To analyze the scenario further, we rewrite the equation to identify critical features such as the vertex, which represents the maximum height, and the points in time when the fireworks reach given heights. The provided function is already in a standard quadratic form, where the coefficient a = -16 signifies the acceleration due to gravity in feet per second squared (assuming standard gravity), the coefficient b = 96 is the initial velocity in feet per second, and c = 6 is the initial height above ground level.

In order to find the height at which the fireworks explode when the fuse is set for 3 seconds, substitute t = 3 into the original height function:

h(3) = -16(3)^2 + 96(3) + 6 = -16(9) + 288 + 6 = -144 + 288 + 6 = 150 ft.

This calculation indicates that the fireworks will explode at a height of 150 feet after 3 seconds.

Next, to determine whether the fireworks reach their maximum height at that time, we analyze the vertex of the parabola. The vertex of a quadratic function h(t) = at^2 + bt + c occurs at t = -b / (2a). Here, a = -16 and b = 96, so:

t = -96 / (2 * -16) = -96 / -32 = 3 seconds.

This indicates that the maximum height occurs at t = 3 seconds, confirming that the firework does indeed explode at its maximum point if the fuse is set for 3 seconds.

Using the vertex form of a quadratic function, h(t) = a(t - h)^2 + k, where (h, k) is the vertex, we find the vertex point directly. Since the vertex occurs at t = 3 seconds and the maximum height is h(3) = 150 ft, the vertex form of the equation becomes:

h(t) = -16(t - 3)^2 + 150.

This form emphasizes how the parabola opens downward with its vertex at (3, 150), consistent with earlier calculations and physical understanding of projectile motion.

In conclusion, this problem exemplifies the quadratic nature of projectile trajectories, illustrating why it is a parabola problem. By analyzing the quadratic function through its vertex and standard form, we can determine key features such as maximum height and specific height at a given time, demonstrating practical applications of algebraic concepts in real-world physics scenarios.

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