Question

A worm is at the bottom of a ravine. Every day it climbs up 5 feet, but every night while it is sleeping, it falls back 3 feet. a) How long will it take the worm to crawl up the ravine if it is 20 ft high? b) What if the ravine is 97 feet high?

Answer 1

a) 9 days

b) 47 days

Explanation:

Given information:

• Distance worm climbs each day = 5 ft
• Distance worm falls back each night = 3 ft

Part (a)

`\begin{array}\cline{1-4} \sf Day & \sf Height\:{(ft) & \sf Night & \sf Height\:(ft) \\\cline{1-4} 1 & 5 & 1 & 5 - 3 = 2\\\cline{1-4} 2 & 2 + 5 = 7 & 2 & 7 - 3 = 4 \\\cline{1-4} 3 & 4 + 5 = 9 & 3 & 9 - 3 = 6 \\\cline{1-4} 4 & 6 + 5 = 11 & 4 & 11 - 3 = 8 \\\cline{1-4} 5 & 8 + 5 = 13 & 5 & 13 - 3 = 10 \\\cline{1-4} 6 & 10 + 5 = 15 & 6 & 15-3=12\\\cline{1-4} 7 & 12 + 5 = 17 & 7 & 17 - 3 = 14 \\\cline{1-4} 8 & 14 + 5 = 19 & 8 & 19-3=16\\\cline{1-4} 9 & 16 + 5 = 21 &&\\\cline{1-4}\end{array}`

On the first day the worm climbs 5 ft, but slips back 3 ft during the first night.

On the second day the worm climbs a further 5 ft, so is now at 7 ft, but slips back 3 ft during the second night. Therefore, it starts the third day at a height of 4 ft.

The progression continues this way so that the worm starts the 9th day at a height of 16 ft. As it climbs 5 feet during the day, it will reach the height of 20 ft during the 9th day, and so will climb out of the ravine before it sleeps for the night.

Therefore, it will take 9 days for the worm to crawl up a 20 ft ravine.

This can be modeled as an arithmetic sequence:

`y=a+(x-1)d`

where:

• y = height of ravine
• x = number of days
• a = initial height after the first day
• d = common difference = 2

Therefore:

`\implies y=5+(x-1)2`

`\implies y=5+2x-2`

`\implies y=3+2x`

Part (b)

To calculate how long it will take the worm to crawl up the ravine if it is 97 ft high, substitute y = 97 into the equation and solve for x:

`\implies 3+2x=97`

`\implies 3+2x-3=97-3`

`\implies2x=94`

`\implies (2x)/(2)=(94)/(2)`

`\implies x=47`

Therefore, it will take 47 days for the worm to crawl up a 97 ft ravine.