Mathematics
Case study:
Cable cars, at hill stations, are major tourist attractions. On a hill station, the length of a cable car ride from the base to the topmost point on the hill is 5000 m. Poles are installed at equal intervals on the way to provide support to the cable on which the car moves.
The distance of the first pole from the base point is 200 m and subsequent poles are installed at equal intervals of 150 m. Further, the distance of the last pole from the top is 300 m.
Based on above information, answer the following questions using Arithmetic Progression:
(i) Find the distance of the 10th pole from the base.
(ii) Find the distance between the 15th pole and 25th pole.
(iii) Find the time taken by the cable car to reach the 15th pole from the top if it is moving at a speed of 5 m/s and coming from the top.
AP
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Answer
(i) Given,
Total cable length = 5000 m
Distance of first pole from base (a) = 200 m
Distance between poles (d) = 150 m
So poles form an A.P. : 200, 350, 500, 650,…
Distance of 10th pole from base,
We know that,
an = a + (n - 1)d
⇒ a10 = 200 + (9) × 150
= 200 + 1350
= 1550 m.
Hence, distance of 10th pole from base = 1550 m.
(ii) Distance between 15th and 25th poles,
We know that,
an = a + (n - 1)d
⇒ a15 = 200 + (14)150
= 200 + 2100
= 2300 m.
⇒ a25 = 200 + (24)150
= 200 + 3600
= 3800 m.
Distance between 15th and 25th poles = 3800 - 2300 = 1500 m.
Hence, distance between 15th and 25th poles is 1500 m.
(iii) Let the total number of poles be n.
The last pole is 300 m from the top (5000 m point), so its distance from the base is : 5000 - 300 = 4700 m.
∴ an = 4700
⇒ a + (n - 1)d = 4700
⇒ 200 + (n - 1)150 = 4700
⇒ 150(n - 1) = 4700 - 200
⇒ 150(n - 1) = 4500
⇒ n - 1 =
⇒ n - 1 = 30
⇒ n = 30 + 1 = 31.
Thus, there are 31 poles in total.
Thus, the 15th pole from the top = (31 - 15 + 1) = 17th pole from the base.
Distance between 31st and 17th pole = (31 - 17)d = 14d = 14 × 150 = 2100 m.
Total distance from top to the 15th pole from the top = 2100 + 300 = 2400 m
Speed = 5 m/s
Time taken = = 480 s or 8 mins.
Hence, time to reach 15th pole from top = 8 minutes.
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