Question

A long-distance runner runs the first kilometre of a race in 3 minutes 45 seconds but finds that his speed drops steadily so that each kilometre takes him 12 seconds more than the preceding one. The time taken to cover the first 12 kilometres is

Answer 1

The time taken to cover the first 12 kilometres by the long-distance runner, whose speed decreases steadily, can be calculated using the sum of an arithmetic progression and equals 3492 seconds.

Step-by-step explanation:

To find the time taken to cover the first 12 kilometres by a long-distance runner with a decreasing speed, we need to use arithmetic progression since each subsequent kilometre takes 12 seconds more than the previous one. The time for the first kilometre is 3 minutes and 45 seconds, which is 225 seconds. Since each km takes 12 more seconds than the previous one, the difference (common difference in an arithmetic progression) is 12 seconds.

The total time taken for the first 12 kilometres can be found using the formula for the sum of the first n terms of an arithmetic progression, which is Sn = ½n(2a + (n - 1)d), where n is the number of terms, a is the first term, and d is the common difference.

Using this formula:

• n = 12
• a = 225 seconds
• d = 12 seconds

The sum S12 is the total time taken for 12 km, which is:

S12 = ½ * 12 * (2 * 225 + (12 - 1) * 12)

After calculating, we get:

S12 = 6 * (450 + 132)

S12 = 6 * 582

S12 = 3492 seconds

Therefore, the runner takes 3492 seconds to cover the first 12 kilometres of the race.