Final answer:
The total distance the ball has traveled, considering the bounces to half the height each time, forms a geometric series. Summing up this series gives us a total travel distance of 120m. The closest whole number provided in the options does not match the computed result.
Step-by-step explanation:
The question asks about the total distance a ball has traveled after being dropped from a certain height and bouncing to half of the previous height each time. Initially, the ball is dropped from 24m. It travels 24m downward to the ground; then it bounces back to half the previous height, which is 12m. The next bounce reaches half of 12m, which is 6m, and so on. This pattern forms a geometric series where each term is half the previous one, and the series can be expressed as 24 + 2*(1/2)*24 + 2*(1/4)*24 + ..., which simplifies to 24 + 24 + 12 + 6 + ...
Summing an infinite series like this can be done using the sum formula for a geometric series: S = a / (1 - r), where 'S' is the sum, 'a' is the first term (24m here), and 'r' is the common ratio (1/2 for the bounces). Given that each bounce goes up and then down for double the distance, we multiply the sum of the upward bounces by 2, except for the initial drop, since it only travels the distance once. Thus, our modified sum formula for the entire travel distance is S = 24 + 2*(24 / (1 - 1/2)), which simplifies to 24 + 2*(24/0.5) = 24 + 2*48 = 24 + 96 = 120m.
So, the correct answer from the given options would be none as the total distance is 120m. The confusion might stem from the options presented, which suggest a whole number answer close to those given, but the actual computation leads to a different result.