In a race, each of the four greyhounds run at its own constant speed. They all start at the same point on a circular track and at 30 seconds into the race, while all are still on their first lap, they have spread out so they are at four corners of a square. How many seconds into the race will it be when they are next at the four corners of a square?

Question

In a race, each of the four greyhounds run at its own constant speed. They all start at the same point on a circular track and at 30 seconds into the race, while all are still on their first lap, they have spread out so they are at four corners of a square. How many seconds into the race will it be when they are next at the four corners of a square?

A. 60
B. 90
C. 150
D. 210
E. 240

Answer 1

clearly, the fastest has run one complete lap, while the slowest has run only 1/4 of a lap.

Then if they started at the eastern side of the track, after 30 seconds they are at the points E,S,W,N in order of speed, fastest to slowest.

After 30 more seconds, they are at E,W,E,W
After another 30 seconds, they are at E,N,W,S

Answer 2

To find the answer to this question, we need to understand the relationship between the greyhounds' positions and the time elapsed in the race.

Let's assume that the circular track has a circumference of 1 unit. Since all four greyhounds start at the same point, they are initially at the same corner of the square.

After 30 seconds, each greyhound will have traveled a distance equal to its constant speed multiplied by the time. Let's represent the speeds of the greyhounds as fractions of the track circumference.

Next, we need to determine the positions of the greyhounds after 30 seconds. Since they have formed a square, each side of the square is equal to the distance traveled by one greyhound.

To form a square, each side of the square should be equal to the diagonal of the square divided by the square root of 2.

So, after 30 seconds, each greyhound is at a distance of (1/4) = 0.25 units away from the starting point.

To determine the time it takes for the greyhounds to return to the corners of a square, we need to find the least common multiple (LCM) of the speeds of the greyhounds.

The speeds of the greyhounds are 1, 2, 3, and 4 (fractions of the track circumference). The LCM of these speeds is 12.

Since the LCM represents the time it takes for all the greyhounds to return to their starting point simultaneously, we need to find a multiple of 12 that is greater than 30 (the initial time of 30 seconds).

The multiples of 12 that are greater than 30 are 36, 48, 60, 72, 84, 96, 108, etc.

We can see that the least multiple of 12 that is greater than 30 is 36. Therefore, the greyhounds will be next at the corners of a square after 36 seconds.

Looking at the answer choices, the closest option to 36 seconds is 60 seconds.

Therefore, the correct answer is A. 60.