To determine which function can be used to represent the height of the T-shirt as a function of time, let's analyze the given details:
1. The T-shirt starts at a height of 8 feet when it leaves the cannon.
2. The T-shirt reaches its maximum height of 24 feet at 1 second.
We know that the path of the T-shirt follows a parabolic trajectory due to the physics of projectile motion. Generally, the formula for the height [tex]\( f(t) \)[/tex] of a projectile at time [tex]\( t \)[/tex] is represented by a quadratic equation in vertex form:
[tex]\[ f(t) = a(t - h)^2 + k \][/tex]
where:
- [tex]\( (h, k) \)[/tex] represents the vertex of the parabola (the point at which the projectile reaches its maximum height).
- [tex]\( a \)[/tex] is a constant that affects the width and direction of the parabola.
Given the vertex (maximum height) at [tex]\( t = 1 \)[/tex] second and the maximum height [tex]\( k = 24 \)[/tex] feet, the vertex form of our equation becomes:
[tex]\[ f(t) = a(t - 1)^2 + 24 \][/tex]
Now, we need to determine the value of [tex]\( a \)[/tex]. We use the fact that at [tex]\( t = 0 \)[/tex] seconds, the height [tex]\( f(0) = 8 \)[/tex] feet.
Substitute [tex]\( t = 0 \)[/tex] and [tex]\( f(0) = 8 \)[/tex] into the vertex form equation:
[tex]\[ 8 = a(0 - 1)^2 + 24 \][/tex]
[tex]\[ 8 = a(1) + 24 \][/tex]
[tex]\[ 8 = a + 24 \][/tex]
Solving for [tex]\( a \)[/tex]:
[tex]\[ a = 8 - 24 \][/tex]
[tex]\[ a = -16 \][/tex]
Now we substitute [tex]\( a = -16 \)[/tex] back into the vertex form equation:
[tex]\[ f(t) = -16(t - 1)^2 + 24 \][/tex]
Thus, the function that represents the height of the T-shirt as a function of time [tex]\( t \)[/tex] in seconds is:
[tex]\[ f(t) = -16(t-1)^2 + 24 \][/tex]
So, the correct function is:
[tex]\[ f(t) = -16(t-1)^2 + 24 \][/tex]
