To solve this problem, we need to analyze each of the given expressions to determine which ones correctly represent the total time of Lina's trip based on the given constraints.
Given:
- For the first 80 miles, Lina travels at a speed of [tex]\( s \)[/tex] miles per hour.
- For the remaining 50 miles, Lina travels 10 miles per hour faster than her speed during the first part of the trip, i.e., at [tex]\( s + 10 \)[/tex] miles per hour.
The total time [tex]\( T \)[/tex] of the trip in hours should be the sum of the time taken for the first 80 miles and the time taken for the remaining 50 miles.
Total Time Calculation:
1. Time for the first 80 miles:
[tex]\[ \text{Time}_1 = \frac{80 \text{ miles}}{s \text{ miles per hour}} = \frac{80}{s} \text{ hours} \][/tex]
2. Time for the remaining 50 miles:
[tex]\[ \text{Time}_2 = \frac{50 \text{ miles}}{(s + 10) \text{ miles per hour}} = \frac{50}{s + 10} \text{ hours} \][/tex]
Therefore, the total time [tex]\( T \)[/tex] is:
[tex]\[ T = \frac{80}{s} + \frac{50}{s + 10} \][/tex]
### Analyzing Each Expression:
1. [tex]\(\frac{80}{s+10}+\frac{s e}{s}\)[/tex]
- The term [tex]\(\frac{s e}{s}\)[/tex] simplifies to [tex]\(e\)[/tex], which does not represent the time relationship given in the problem.
- Incorrect Expression
2. [tex]\(\frac{80}{x}+\frac{50}{x+10}\)[/tex]
- This matches our derived expression with [tex]\( x \)[/tex] representing [tex]\( s \)[/tex].
- Correct Expression
3. [tex]\(\frac{130 a-809}{s(a-10)}\)[/tex]
- This form does not relate to the formula for calculating time based on speeds and distances given.
- Incorrect Expression
4. [tex]\(\frac{130 t+800}{a(s+10)}\)[/tex]
- This form and the variables do not conform to the correct expression derived.
- Incorrect Expression
5. [tex]\(\frac{130}{3(0+10)}\)[/tex]
- This is a numerical expression unrelated to the variables and relationships provided in the problem.
- Incorrect Expression
### Conclusion:
The only correct expression representing the total time of Lina's trip in hours is:
[tex]\[ \frac{80}{x} + \frac{50}{x + 10} \][/tex]
Hence, the correct answers are:
[tex]\[ \boxed{\frac{80}{x} + \frac{50}{x + 10}} \][/tex]
