Question

In order to organise, Annual sports Day, a school prepared an eight-lane running track with an integrated football field inside the track area as shown below: The length of the innermost lane of the track is 400 m and each subsequent lane is 7.6 m longer than the preceding lane.

Based on the given information, answer the following questions, using the concept of arithmetic progression.

1. What is the length of the 6^th lane?   (1)
2. How long is the 8^th lane than that of 4^th lane?   (1)
3. 1. While practicing for a race, a student took one round each in first six lanes. Find the total distance covered by the student.  (2)
                                         OR
   2. A student took one round each in lane 4 to lane 8. Find the total distance covered by the student. (2)

Answer 1

a = 400 m

d  = 7.6 m

i. a_n ​= a + (n − 1)d

a₆ ​= 400 + (6 − 1) × 7.6

= 400 + 5 × 7.6

= 400 + 38

= 438 m

So, the length of the 6^th lane is 438 m.

ii. For the 8^th lane (n = 8):

a₈ ​= 400 + (8 − 1) × 7.6

= 400 + 7 × 7.6

= 400 + 53.2

= 453.2 m

For the 4^th lane (n = 4):

a₄ ​= 400 + (4 − 1) × 7.6

= 400 + 3 × 7.6

= 400 + 22.8

= 422.8 m

Difference = a₈ − a₄

= 453.2 − 422.8 

= 30.4 m

So, the 8^th lane is 30.4 m longer than the 4^th lane.

iii. a. S_n = `n/2[2a + (n - 1)d]`

S₆ = `6/2[2(400) + (6 - 1) xx 7.6]`

= 3[800 + 5 × 7.6]

= 3[800 + 38]

= 3 × 838

= 2514 m

So, the total distance covered by the student is 2514 m.

OR

b. a = a₄ = 422.8 m

1 = a₈ = 453.2 m

Sum = `n/2 [a + a_8]`

= `5/2 [422.8 + 453.2]`

= `5/2 [876]`

= 5 × 438

= 2190 m

So, the total distance covered by the student is 2190 m.