To determine the most appropriate scale for the vertical axis that maximizes the difference in the heights of the histogram bars, let's analyze the given information:
1. Number of People in Each Salary Range:
- [tex]$0-\$[/tex]19,999[tex]$: 40 people
- $[/tex]20,000-\[tex]$39,999$[/tex]: 30 people
- [tex]$40,000-\$[/tex]59,999[tex]$: 35 people
2. Differences in Counts Between Consecutive Bars:
- Difference between $[/tex]0-\[tex]$19,999$[/tex] and [tex]$20,000-\$[/tex]39,999[tex]$:
\[
40 - 30 = 10
\]
- Difference between $[/tex]20,000-\[tex]$39,999$[/tex] and [tex]$40,000-\$[/tex]59,999[tex]$:
\[
30 - 35 = 5
\]
- Difference between $[/tex]0-\[tex]$19,999$[/tex] and [tex]$40,000-\$[/tex]59,999[tex]$:
\[
40 - 35 = 5
\]
3. Maximum Count:
- The maximum count across all salary ranges is 40 people.
This means that our vertical axis must accommodate at least up to 40.
4. Choose Vertical Scale to Maximize Differences:
The largest difference is between $[/tex]0-\[tex]$19,999$[/tex] and [tex]$20,000-\$[/tex]39,999$, which is 10 people.
Now, considering the given options:
- [tex]\(0-50\)[/tex]: This scale covers all values up to the maximum count (40 people) and shows the differences effectively.
- [tex]\(0-40\)[/tex]: This scale exactly covers the maximum count but doesn't offer additional space for clarity.
- [tex]\(10-50\)[/tex]: This scale is not suitable because it does not start from 0, which may misrepresent the actual bar heights.
- [tex]\(25-40\)[/tex]: This scale is also not suitable because it does not start from 0 and does not represent the smaller counts properly.
Therefore, the most appropriate scale for the vertical axis that maximizes the difference in the heights of the bars is [tex]\(0-50\)[/tex].
[tex]\[
\boxed{0-50}
\][/tex]
