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Questions and Answers
What is the equation describing the height of a ball shot into the air at time t?
When does the ball reach its highest point after being shot into the air?
What is the maximum height of the ball when it reaches its peak?
How can we find the exact location of the maximum height using the completing the square method?
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What is the derivative of the height function h(t) with respect to time t?
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What does a negative value for the derivative of the height function imply about the ball's motion?
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Study Notes
Word problems involving quadratic equations require students to analyze situations, set up equations, and manipulate algebraic expressions to find the desired values. These problems often revolve around scenarios where an object moves along a path described by a quadratic function, such as projectiles, rolling balls, or elastic collisions. Here are some strategies and examples of solving word problems that involve quadratic equations.
Finding Maximum Height:
Consider the example of a ball being shot into the air, whose height h at time t is described by the equation h(t) = -16t² + 40t + 1.5. By analyzing the quadratic function h, we find that the maximum height occurs when the derivative of h is zero. Taking the derivative of h with respect to t, we get dh/dt = -32t + 40. Setting this equal to zero and solving for t, we find that the ball reaches its highest point at t = 5/2. Hence, the maximum height is h(5/2) = 10 + 30/16 = 40/8 = 5 units.
Using Completing the Square Method:
Alternatively, we can use the completing the square method to find the exact location of the maximum height. Rewriting the quadratic function as h(t) = (-16/4)t² + (40/16)t + (1.5/16), we recognize it as a perfect square trinomial. By taking the square root of the coefficient of t², we get √(-16/4) = √4 or 2. Adding this square root to each term and multiplying by the constant (1/16), we obtain: h(t) = (t + 2)² - (3/4). The maximum occurs at t = -2, yielding the same result as before.
In summary, solving word problems involving quadratic equations requires a solid understanding of quadratic functions, including knowledge of their properties, methods like the quadratic formula or completing the square, and problem-solving skills. Practice and patience are crucial in developing the ability to tackle various quadratic scenarios effectively.
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Description
Explore strategies and examples for solving word problems involving quadratic equations. Learn how to find the maximum height of objects, analyze scenarios described by quadratic functions, and use methods like the derivative and completing the square to solve quadratic word problems effectively.
