Final answer:
Given the initial speed and deceleration, the time to stop for the runner is approximately 16 seconds, which indicates the runner would cross the finish line before stopping. Options provided in the question are therefore unreasonable.
Step-by-step explanation:
A runner approaches the finish line and is 75 m away; her speed at this position is 8 m/s. She accelerates opposite to the motion at this point at 0.5 m/s². To determine how long it takes her to cross the finish line from 75 m away, we use the equation of motion:
s = ut + ½(at²)
Where s is the distance to the finish line (75 m), u is the initial velocity (8 m/s), a is the acceleration (-0.5 m/s² - negative because it's opposite to the motion), and t is the time we are solving for.
Rearranging the equation for time, we get a quadratic equation:
½(at²) + ut - s = 0
Using the quadratic formula, we find that the time t comes out as either positive or negative; the negative value is discarded because time cannot be negative in this context. The positive solution is the time it takes for the runner to stop, given the deceleration. Plugging in the values, we find that the time taken to stop is approximately 16 s, which is more than 15.0 s, making option (b) incorrect as it is not enough time for the runner to stop.
Also, s is 75 m and we need to find the time till the runner stops, not necessarily crosses the finish line, and if the time to stop is more than 15 s the runner will overshoot the finish line before coming to a stop.
The result of around 16 s to stop indicates that the runner will actually pass the finish line before coming to a stop. Considering she is decelerating, she will definitely cross the finish line before stopping - implying that all options given are unreasonable as they suggest it would take longer to reach the finish line than it would take to stop completely.
However, since none of the options match this analysis, we cannot conclude any of them as reasonable or correct in this context.