Final answer:
The winner in the fourth race beats the runner who comes third by 16 seconds.
Step-by-step explanation:
To find out by how many seconds the winner in the fourth race beats the runner who comes third, we first need to determine the speeds of each runner. Let's assign the speed of runner A as 'a', the speed of runner B as 'b', and the speed of runner C as 'c'.
In race 1, A gives B a 10m head start, and the race is a tie. This means that A runs 100m in the same time it takes B to run 110m:
a = (100m) / t, b = (110m) / t
From this equation, we find that a = 100/t and b = 110/t.
In race 2, B gives C a 15m head start, and the race is a tie. This means that B runs 100m in the same time it takes C to run 115m:
b = (100m) / t, c = (115m) / t
From this equation, we find that b = 100/t and c = 115/t.
In race 3, C gives A a 20m head start, and the race is a tie. This means that C runs 100m in the same time it takes A to run 120m:
c = (100m) / t, a = (120m) / t
From this equation, we find that c = 100/t and a = 120/t.
Now we can determine the speeds of each runner in terms of t: a = 100/t, b = 100/t, c = 100/t.
In race 4, A, B, and C race together. Since they all run at the same speed, the winner beats the runner who comes third by the same margin in each race. Therefore, the winner in the fourth race beats the runner who comes third by (100 + 100 + 100) / t - 100 / t = 200 / t seconds.
To find out how many seconds this is, we need to find the value of t. To do this, we can use the information given in the question:
a = 100/t, b = 110/t, c = 120/t
Since races 1, 2, and 3 are all ties, we can set a = b = c and solve for t:
100/t = 110/t = 120/t
From this equation, we find that t = 12.5 seconds.
Substituting this value into our earlier equation, we find that the winner in the fourth race beats the runner who comes third by 200/12.5 = 16 seconds (to 3 s.f.).