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

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

. 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.

Answer 2

To rewrite the function h(t) in the form h(t) = a(t-h)^2 + k, we need to complete the square.

Given: h(t) = -4t^2 + 24t

Step 1: Factor out the common factor of -4 from the terms involving t:
h(t) = -4(t^2 - 6t)

Step 2: To complete the square, we need to take half of the coefficient of t (-6) and square it:
h(t) = -4(t^2 - 6t + (-6/2)^2)

Simplifying the equation in the parentheses:
h(t) = -4(t^2 - 6t + 9) [(-6/2)^2 = 9]

Step 3: Rewrite the equation, factoring the perfect square trinomial:
h(t) = -4(t - 3)^2 + 36

Now we have h(t) in the desired form: h(t) = a(t-h)^2 + k, where a = -4, h = 3, and k = 36.

The maximum height of the projectile can be determined by looking at the value of k. In this case, the maximum height is given by k, which is 36.

Answer 3

To rewrite the given function in the form h(t) = a(t-h)^2 + k, we need to complete the square.

Given function: h(t) = -4t^2 + 24t

Step 1: Factor out the common term from the equation.
h(t) = -4(t^2 - 6t)

Step 2: To complete the square, we need to take half of the coefficient of 't', square it, and add it inside the parentheses.
h(t) = -4(t^2 - 6t + 9 - 9)

Step 3: Rearrange the terms inside the parentheses.
h(t) = -4((t^2 - 6t + 9) - 9)

Step 4: Factorize the perfect square trinomial inside the parentheses.
h(t) = -4((t - 3)^2 - 9)

Step 5: Expand the equation.
h(t) = -4(t - 3)^2 + 36

Now we have rewritten the function in the desired form h(t) = a(t-h)^2 + k, where a = -4, h = 3, and k = 36.

The maximum height of the projectile is given by the value of k, which is 36. So, the maximum height of the projectile is 36 feet.