|
Question 966937: in a computer game room there is a mechanical grasshopper in a glasscage. The grasshopper is placed 42 inches from the door of the cage. each time the button is pressed, the grasshopper jumps one-third the remaining distance to the door. after how may jumps does the grasshopper reach the door?
how far does the grasshopper travel after the fifth jump?
how far from the door is the grasshopper after the third jump?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the grass hopper will never reach the door.
each time the grasshopper jumps 1/3 of the remaining distance to the door.
after the first jump, the remaining distance is 42 * 2/3.
after the second jump, the remaining distance is 42 * 2/3 * 2/3 = 42 * (2/3)^2.
after the third jump, the remaining distance is 42 * 2/3 * 2/3 * 2/3 = 42 * (2/3)^3.
fourth = 42 * (2/3)^4
fifth = 42 * (2/3)^5
one thousandth = 42 * (2/3)^1000
the remaining distance never gets to zero because the remaining distance is always 42 * (2/3)^n, where (2/3)^n is a positive number, no matter how small.
the remaining distances for the first 5 hopes are shown below:
42 * (2/3)^1 = 28
42 * (2/3)^2 = 18.67
42 * (2/3)^3 = 12.44
42 * (2/3)^4 = 8.30
42 * (2/5)^5 = 5.53
the grasshopper will have jumped one half of the remaining distance to get to that remaining distance.
for example:
for the first hop, the grasshopper hopped 1/3 * 42 = 14.
the remaining distance is 2/3 * 42 = 28.
14 is equal to 1/2 of 28.
these are the remaining distances for the first 5 hops.
42 * (2/3)^1 = 28
42 * (2/3)^2 = 18.67
42 * (2/3)^3 = 12.44
42 * (2/3)^4 = 8.30
42 * (2/5)^5 = 5.53
the grasshopper hopped the following distances to get to each remaining distance.
42 * (2/3)^1 = 28 means the grasshopper hopped 14 for the first hop.
42 * (2/3)^2 = 18.67 means the grasshopper hopped 9.335 for the second hop.
42 * (2/3)^3 = 12.44 means the grasswhopper hopped 6.22 for the third hop.
42 * (2/3)^4 = 8.30 means the grasshopper hopped 4.15 for the fourth hop.
42 * (2/5)^5 = 5.53 means the grasshopper hopped 2.765 for the fifth hop.
the grasshopper had hopped 29.555 inches total for the first 3 hops which means the grasshopper was 42 - 29.555 = 12.445 inches from the door.
the grasshopper had hopped 36.47 inches total for the first 5 hops.
|
|
|
| |
