Between the mountain top and foot of 3600 m two cable cars of A and B is scheduled to be operated as graph below.
It takes 32 minutes for descent and 40 minutes for ascent for both cable cars and they advance with a fixed speed.
Answer the following questions.
(1) As scheduled, A and B left at the same time.
How many meters from the foot do 2 cable cars pass each other first?
(2) Since it took time for checking, B was late to start descending for 10 minutes.
How many meters from the foot do 2 cable cars pass each other first in this case?
(3) The place where 2 cable cars pass each other was the spot in the middle of the top and foot of the mountain.
Answer which cable car was late to start in this case.
How many minutes was it late for leaving?
Answer
(1) 1600 m
(2) 2100 m
(3) B delayed 4 minutes
Solution
(1) The time ratio of ascent and descent is 40 : 32 = 5 : 4.
Because the speed ratio is to be inverse ratio of the time ratio and ascent : descent = 4 : 5.
Since the moving distance is proportional to the speed, the distance from the foot where 2 cable cars passed each other is 3600 × 4/(4+5 ) = 1600 m.
(2) The upper triangle and the lower triangle are homothetic in a figure below and the homothetic ratio is (40- 10) : 42 = 5 : 7.
Accordingly the distance from the foot is 3600 × 7/(5 + 7) =2100 m.
(3) According to the result of (1) and (2), when departure of B is behind schedule, it is found out that the distance from the foot becomes longer than the case of leaving as scheduled.
Therefore it's B that was behind the schedule in case the place passing each other is the middle of the top and foot.
It takes 32 minutes for descent and 40 minutes for ascent for both cable cars and they advance with a fixed speed.
Answer the following questions.
(1) As scheduled, A and B left at the same time.
How many meters from the foot do 2 cable cars pass each other first?
(2) Since it took time for checking, B was late to start descending for 10 minutes.
How many meters from the foot do 2 cable cars pass each other first in this case?
(3) The place where 2 cable cars pass each other was the spot in the middle of the top and foot of the mountain.
Answer which cable car was late to start in this case.
How many minutes was it late for leaving?
Answer
(1) 1600 m
(2) 2100 m
(3) B delayed 4 minutes
Solution
(1) The time ratio of ascent and descent is 40 : 32 = 5 : 4.
Because the speed ratio is to be inverse ratio of the time ratio and ascent : descent = 4 : 5.
Since the moving distance is proportional to the speed, the distance from the foot where 2 cable cars passed each other is 3600 × 4/(4+5 ) = 1600 m.
(2) The upper triangle and the lower triangle are homothetic in a figure below and the homothetic ratio is (40- 10) : 42 = 5 : 7.
Accordingly the distance from the foot is 3600 × 7/(5 + 7) =2100 m.
(3) According to the result of (1) and (2), when departure of B is behind schedule, it is found out that the distance from the foot becomes longer than the case of leaving as scheduled.
Therefore it's B that was behind the schedule in case the place passing each other is the middle of the top and foot.
It is assumed that B was late to start ① minutes.
In the figure below, the upper triangle and the lower triangle become congruent.
40 - ① = 32 + ①, then ② = 8.
Therefore ① = 4 and B was late for 4 minutes.
In the figure below, the upper triangle and the lower triangle become congruent.
40 - ① = 32 + ①, then ② = 8.
Therefore ① = 4 and B was late for 4 minutes.
