To determine the reasonable domain for the function \( h(t) = 160t - 16t^2 \), we need to consider the physical context of the problem. The function describes the height of a ball thrown upward, which implies that:
- Time \( t \) cannot be negative since we start measuring time from the moment the ball is thrown (which is at \( t = 0 \)).
- We also need to consider when the ball hits the ground (i.e., when \( h = 0 \)), which occurs at a certain positive time.
To find when the ball hits the ground, we set \( h(t) = 0 \):
\[ 160t - 16t^2 = 0 \] Factoring out \( t \):
\[ t(160 - 16t) = 0 \] This gives us two solutions:
- \( t = 0 \) (the starting time)
- \( 160 - 16t = 0 \) → \( 16t = 160 \) → \( t = 10 \)
Thus, the ball will be in the air from \( t = 0 \) to \( t = 10 \) seconds. Therefore, the reasonable domain of the function is the interval:
\[ 0 \leq t \leq 10 \]
This corresponds to option A: {x: 0 ≤ x ≤ 10}.