Find the maximum height of the projectile modeled by h(t) = -16t^2 + 32t + 48. (

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Question: Find the maximum height of the projectile modeled by h(t) = -16t^2 + 32t + 48. (2020)

Options:

Correct Answer: 48

Exam Year: 2020

Solution:

The maximum occurs at t = -b/(2a) = -32/(2*-16) = 1. h(1) = 64.

Find the maximum height of the projectile modeled by h(t) = -16t^2 + 32t + 48. (2020)

Find the maximum height of the projectile modeled by h(t) = -16t^2 + 32t + 48. (2020)

Step 1: Identify the equation of the projectile's height, which is h(t) = -16t^2 + 32t + 48.
Step 2: Recognize that this is a quadratic equation in the form h(t) = at^2 + bt + c, where a = -16, b = 32, and c = 48.
Step 3: To find the maximum height, use the formula for the time at which the maximum occurs: t = -b/(2a).
Step 4: Substitute the values of a and b into the formula: t = -32/(2 * -16).
Step 5: Calculate the denominator: 2 * -16 = -32, so t = -32 / -32 = 1.
Step 6: Now that we have t = 1, substitute this value back into the original height equation to find the maximum height: h(1) = -16(1)^2 + 32(1) + 48.
Step 7: Calculate h(1): h(1) = -16(1) + 32 + 48 = -16 + 32 + 48.
Step 8: Simplify the calculation: -16 + 32 = 16, and then 16 + 48 = 64.
Step 9: Therefore, the maximum height of the projectile is 64.

Quadratic Functions – Understanding the properties of quadratic functions, including how to find the vertex, which represents the maximum or minimum point.
Vertex Formula – Using the vertex formula t = -b/(2a) to find the time at which the maximum height occurs in a projectile motion equation.
Evaluating Functions – Substituting the value of t back into the height function h(t) to find the maximum height.