Given that the quadratic function modeling the height of a ball over time is symmetric about the line [tex]\(t = 2.5\)[/tex], let's analyze the statements to determine the correct one.
The symmetry of the quadratic function means that the height of the ball at times [tex]\(t\)[/tex] and [tex]\( (5 - t) \)[/tex] will be the same because [tex]\(t = 2.5\)[/tex] is the midpoint.
Let's examine each statement:
A. The height of the ball is the same after 0.5 seconds and 5.5 seconds.
- Midpoints: [tex]\(\frac{0.5 + 5.5}{2} = 3\)[/tex]. This is not symmetric about [tex]\(t = 2.5\)[/tex].
- False.
B. The height of the ball is the same after 1.5 seconds and 3.5 seconds.
- Midpoints: [tex]\(\frac{1.5 + 3.5}{2} = 2.5\)[/tex]. This is symmetric about [tex]\(t = 2.5\)[/tex].
- True.
C. The height of the ball is the same after 1 second and 3 seconds.
- Midpoints: [tex]\(\frac{1 + 3}{2} = 2\)[/tex]. This is not symmetric about [tex]\(t = 2.5\)[/tex].
- False.
D. The height of the ball is the same after 0 seconds and 4 seconds.
- Midpoints: [tex]\(\frac{0 + 4}{2} = 2\)[/tex]. This is not symmetric about [tex]\(t = 2.5\)[/tex].
- False.
Based on this analysis, statement B is the only one where the midpoints are equal to 2.5, making it symmetric about [tex]\(t = 2.5\)[/tex].
Thus, the correct answer is:
[tex]\[ \boxed{B} \][/tex]
