Let's carefully analyze the problem step by step and determine which solution can be eliminated and why.
1. Understand the Equation:
The height of the football at time [tex]\( t \)[/tex] is modeled by the quadratic equation:
[tex]\[
h(t) = -16t^2 + 40t + 7
\][/tex]
We are interested in finding when the football hits the ground, which happens when [tex]\( h(t) = 0 \)[/tex].
2. Solve for [tex]\( t \)[/tex] when [tex]\( h(t) = 0 \)[/tex]:
Setting [tex]\( h(t) \)[/tex] to zero, we get:
[tex]\[
-16t^2 + 40t + 7 = 0
\][/tex]
Solving this quadratic equation, the roots (or solutions) are given as [tex]\( t = -0.2 \)[/tex] seconds and [tex]\( t = 2.7 \)[/tex] seconds.
3. Evaluating the Solutions:
- The first solution is [tex]\( t = -0.2 \)[/tex] seconds.
- The second solution is [tex]\( t = 2.7 \)[/tex] seconds.
4. Physical Meaning of Time:
In the context of time, [tex]\( t \)[/tex] represents how long after the pass was thrown that the football hits the ground. Time cannot be negative because it would not make physical sense to have an event occur before the pass was thrown.
5. Elimination of the Solution:
- The negative value [tex]\( t = -0.2 \)[/tex] seconds is not physically meaningful because time cannot be a negative quantity. Therefore, this solution can be eliminated.
- The positive value [tex]\( t = 2.7 \)[/tex] seconds is a valid solution as it represents a feasible duration of time after the pass was thrown when the football hits the ground.
Thus, the correct solution to be eliminated is [tex]\( t = -0.2 \)[/tex] seconds, and the reason is as follows:
[tex]\[
\text{The solution } -0.2 \text{ s can be eliminated because time cannot be a negative value.}
\][/tex]
Therefore, the correct choice is:
[tex]\[
\boxed{\text{The solution } -0.2 \text{ s can be eliminated because time cannot be a negative value.}}
\][/tex]
