Question

Cable cars at hill stations are one of the major tourist attractions. On a hill station, the length of the cable car ride from the base point to the top-most point on the hill is 5000 m. Poles are installed at equal intervals on the way to provide support to the cables on which the car moves. The distance of the first pole from the base point is 200 m and subsequent poles are installed at an equal interval of 150 m. Further, the distance of the last pole from the top is 300 m.

Based on the above information, answer the following questions using arithmetic progression:

1. Find the distance of the 10^th pole from the base. (1)
2. Find the distance between the 15^th pole and the 25^th pole. (1)
3. 1. Find the time taken by the cable car to reach the 15^th pole from the top if it is moving at the speed of 5 m/sec and coming from the top. (2)
      OR
   2. Find the total number of poles installed along the entire journey. (2)

Answer 1

i.

Distance of P₁ from base = 200 m

Distance of P₂ from base = 200 + 150

= 350 m

A.P. 200, 350, 500

a = 200, d = 150

Distance of the 10^th pole from the base = a₁₀ = a + 9d  ...[∵ T_n = a + (n − 1)d]

= 200 + 9 × 150 

= 200 + 1350

= 1550 m

ii. Distance between the 15^th pole and the 25^th pole = a₂₅ − a₁₅

= (a + 24d) – (a + 14d)

= a + 24d – a – 14d

= 10 × d

= 10 × 150

= 1500 m

iii. a. Distance of the 15^th pole from the top = 300, 450

= 300 + (15 – 1) × 150

= 300 + 14 × 150

= 300 + 2100

= 2400 m

Time = `"Distance"/"Speed"`

= `2400/5`

= 480 sec.

or `480/60`

= 8 min

As 8 minute or 480 seconds.

OR

b. Distance of P_n from base = 5000 − 300

= 4700

a_n = a + (n − 1)d

4700 = 200 + (n – 1) × 150

4700 – 200 = (n – 1) × 150

`4500/150` = n – 1

30 = n – 1

n = 31 poles