To determine which mathematical sentence correctly represents the situation described in the problem, let's go through each option and analyze it carefully:
The problem states: Dan was riding in a bicycle race, and after the first five hours of racing, he had ridden more than 175 km. We need to determine what mathematical sentence can describe his average speed, [tex]\( s \)[/tex].
1. Option A: [tex]\( s + 5 > 175 \)[/tex]
This option suggests that Dan's average speed plus 5 is greater than 175. This does not make sense for our problem because adding time to speed does not help us find a relationship involving distance.
2. Option B: [tex]\( 5s < 175 \)[/tex]
This option states that five times the average speed is less than 175, which contradicts our problem statement of riding more than 175 km in five hours.
3. Option C: [tex]\( 5s > 175 \)[/tex]
This option indicates that five times the average speed is greater than 175, which aligns with the problem since it states that Dan rode more than 175 km in five hours. Mathematically, this is expressed as:
[tex]\[
\text{Distance} = \text{Speed} \times \text{Time}
\][/tex]
Given that the distance (175 km) is more than [tex]\( 5s \)[/tex], we can write:
[tex]\[
5s > 175
\][/tex]
This correctly represents the situation.
4. Option D: [tex]\( 5^5 = 175 \)[/tex]
This option states that 5 raised to the power of 5 equals 175, which is a specific numerical operation unrelated to the context of speed, time, and distance.
5. Option E: [tex]\( \frac{\theta}{5} \geq 175 \)[/tex]
This option uses an unfamiliar variable [tex]\( \theta \)[/tex] and states that this variable divided by 5 is greater than or equal to 175. This does not match the problem's variables or context.
6. Option F: [tex]\( 175 - 5s = 0 \)[/tex]
This option states that 175 minus five times the speed equals zero, which suggests [tex]\( 5s = 175 \)[/tex]. This only supports equality, not the "more than" condition given in the problem statement.
Thus, the correct mathematical sentence that accurately translates the problem is:
Option C: [tex]\( 5s > 175 \)[/tex]
