The vertical height in feet of a projectile on a planet in our solar system at a given time t in seconds is represented by the function h(t)=−4t2+24t

Question

. Re-write h(t) in the form \large h\left(t\right)=a\left(t-h\right)^2+k and determine the maximum height of the projectile. Show all work to receive credit for this question.

Answer 1

To re-write h(t) in the desired form, we need to complete the square.

h(t) = -4t^2 + 24t
h(t) = -4(t^2 - 6t)

To complete the square, we need to add and subtract (6/2)^2 = 9 inside the parentheses.

h(t) = -4(t^2 - 6t + 9 - 9)
h(t) = -4((t-3)^2 - 9)

Simplifying:

h(t) = -4(t-3)^2 + 36

Now we have the desired form with a = -4, h = 3, and k = 36.

To determine the maximum height, we note that the vertex of the parabola is located at the point (h, k). Therefore, the maximum height is k = 36, which occurs when t = 3 seconds.