Find the maximum height of the projectile modeled by h(t) = -16t^2 + 64t + 48. (

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Question: Find the maximum height of the projectile modeled by h(t) = -16t^2 + 64t + 48. (2020)

Options:

Correct Answer: 64

Solution:

The maximum occurs at t = -b/(2a) = 64/(2*16) = 2. h(2) = -16(2^2) + 64(2) + 48 = 80.

Find the maximum height of the projectile modeled by h(t) = -16t^2 + 64t + 48. (2020)

Find the maximum height of the projectile modeled by h(t) = -16t^2 + 64t + 48. (2020)

Step 1: Identify the equation of the projectile's height, which is h(t) = -16t^2 + 64t + 48.
Step 2: Recognize that this is a quadratic equation in the form h(t) = at^2 + bt + c, where a = -16, b = 64, 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 b and a into the formula: t = -64/(2 * -16).
Step 5: Calculate the denominator: 2 * -16 = -32, so t = -64 / -32 = 2.
Step 6: Now, substitute t = 2 back into the height equation to find the maximum height: h(2) = -16(2^2) + 64(2) + 48.
Step 7: Calculate 2^2, which is 4, then multiply: -16 * 4 = -64.
Step 8: Calculate 64 * 2, which is 128.
Step 9: Now, combine the results: h(2) = -64 + 128 + 48.
Step 10: Add -64 and 128 to get 64, then add 48 to get the final maximum height: 64 + 48 = 112.

Quadratic Functions – Understanding the properties of quadratic functions, including finding the vertex which represents the maximum or minimum value.
Projectile Motion – Applying the quadratic equation to model the height of a projectile over time.
Vertex Formula – Using the vertex formula t = -b/(2a) to find the time at which the maximum height occurs.