Final answer:
To determine by how much time A will beat B, calculate the time it takes for B to reach the point of injury and the time taken to complete the race afterwards with the reduced speed. A's time remains constant. The closest correct answer, based on calculations, is approximately 72.22 seconds, which is not an available option.
Step-by-step explanation:
The subject of this question is Mathematics, specifically involving concepts related to speed and velocity. To solve this problem, we need to compute the respective times it takes for runner A and runner B to finish the race, including the adjustment for B's injury and reduced speed.
Initially, we know that A beats B by 100 m or 10 seconds in a 1000 m race. This implies that A's speed is 100 m / 10 s = 10 m/s, while B's speed is (1000 m - 100 m) / (1000 m / 10 m/s) = 900 m / 100 s = 9 m/s.
If B gets injured after running 50 m less than half the race, B would have run 450 m (since half of 1000 m is 500 m, and 500 m - 50 m is 450 m). At the speed of 9 m/s, B takes 450 m / 9 m/s = 50 seconds to reach the point of injury.
After the injury, B's speed is halved to 4.5 m/s, and he has 550 m left to run. This will take 550 m / 4.5 m/s = 122.22 seconds. So, B's total time to finish the race after getting injured is 50 s (before injury) + 122.22 s (after injury) = 172.22 seconds.
Since A runs at a constant speed of 10 m/s, A finishes the race in 1000 m / 10 m/s = 100 seconds.
The time by which A beats B is B's total time minus A's time, which is 172.22 s - 100 s = 72.22 seconds. Since this is not an option provided in the problem, it might be that the question contains a typo or the answer choices are incorrect. The correct answer is closest to 72.22 seconds, but not listed as an option.