Let's break down the problem step by step.
1. Understand the function:
The function [tex]\( f(x) = -4.9x^2 + 9.8x + 0.6 \)[/tex] represents the height [tex]\( y \)[/tex] of the water balloon at any given time [tex]\( x \)[/tex] after it is launched.
2. Initial height:
The water balloon is launched from a stand 0.6 meters tall. Therefore, at [tex]\( x = 0 \)[/tex] (the moment of launch), the height [tex]\( y \)[/tex] is [tex]\( f(0) = 0.6 \)[/tex]. This means the initial height of the water balloon is 0.6 meters.
3. Maximum height:
The task specifically states that the maximum height of the water balloon is [tex]\( y \leq 5.5 \)[/tex] meters. This indicates the highest point the water balloon can reach during its trajectory.
4. Reasonable range vs. mathematical range:
- The mathematical range includes all possible values of [tex]\( y \)[/tex] that the function can take.
- The reasonable range, on the other hand, is the practical range of heights that make sense for this problem. Realistically, the height cannot go below the initial height of the stand (0.6 meters) and cannot surpass the maximum height given (5.5 meters).
5. Conclusion for the reasonable range:
Given that the water balloon starts at 0.6 meters and can reach up to 5.5 meters, the reasonable range for the height of the water balloon is:
[tex]\[
0.6 \leq y \leq 5.5
\][/tex]
Therefore, the reasonable range is limited to [tex]\( 0.6 \leq y \leq 5.5 \)[/tex].
The correct answer is:
The reasonable range is limited to [tex]\(0.6 \leq y \leq 5.5\)[/tex].
