To determine the correct function that models the height of the baseball, we need to consider the properties of a quadratic function given in the form [tex]\( h(t) = a(t-h)^2 + k \)[/tex]. Here, [tex]\((h, k)\)[/tex] is the vertex of the parabola, which represents the maximum height of the projectile.
Given the problem:
- The baseball is initially hit from a height of [tex]\(3\)[/tex] feet.
- The baseball reaches a maximum height of [tex]\(403\)[/tex] feet.
Based on these details, let's evaluate the options provided:
Option A: [tex]\( h(t) = -16(t-3)^2 + 403 \)[/tex]
This function suggests:
- The horizontal shift [tex]\((t-3)\)[/tex] means the vertex occurs at [tex]\(t=3\)[/tex] seconds.
- The maximum height [tex]\(k\)[/tex] is [tex]\(403\)[/tex] feet.
However, the initial height is not correctly represented when [tex]\(t=0\)[/tex]. Therefore, this is not the correct option.
Option B: [tex]\( h(t) = -16(t-5)^2 + 3 \)[/tex]
This function suggests:
- The horizontal shift [tex]\((t-5)\)[/tex] means the vertex occurs at [tex]\(t=5\)[/tex] seconds.
- The initial height [tex]\(k\)[/tex] is [tex]\(3\)[/tex] feet.
However, the maximum height here is just [tex]\(3\)[/tex] feet, which doesn't match the given maximum height of [tex]\(403\)[/tex] feet. Thus, B is not correct.
Option C: [tex]\( h(t) = -16(t-5)^2 + 403 \)[/tex]
This function suggests:
- The vertex occurs at [tex]\(t=5\)[/tex] seconds where the maximum height is [tex]\(403\)[/tex] feet.
- When [tex]\(t=0\)[/tex], substituting into the function gives us:
[tex]\[
h(0) = -16(0-5)^2 + 403
\][/tex]
[tex]\[
= -16(25) + 403
\][/tex]
[tex]\[
= -400 + 403
\][/tex]
[tex]\[
= 3
\][/tex]
The initial height correctly matches [tex]\(3\)[/tex] feet, making this the correct function.
Option D: [tex]\( h(t) = -16(t-403)^2 + 3 \)[/tex]
This function suggests:
- The vertex occurs at [tex]\(t=403\)[/tex] seconds which doesn't make sense in the context of typical baseball flight times.
- The initial height [tex]\(k\)[/tex] is incorrectly represented if we consider real-world scenarios.
Given these evaluations, the correct function that models the situation accurately is:
[tex]\( \boxed{C} \)[/tex]
