1. 20 rabbits are left alone on a small island off of the coast of Japan. The number of rabbits on the island after 1 year is 40. After 2 years there are 80 rabbits, and after 3 years there are 160 rabbits.
a. Assume that this pattern continues. Make a table showing how many rabbits there will be on the island for the first 7 years. Include 0 years, which is the initial number of rabbits.
b. What type of relationship does the table suggest? Explain why.
c. Write an equation for in terms of , where represents the number of rabbits and represents the number of years.
d. Explain what each part of the equation tells us about how the number of rabbits changes.
e. Predict how many rabbits there will be after 20 years. Does this answer make sense? Why or why not?
Answers
Answer —
Given information –
- There are 20 rabbits in the initial period.
- In the first year, the number of rabbits increases to 40.
- In the second year, the number of rabbits increases to 80.
- In the third year, the number of rabbits increases to 160.
So, here we observe that every year the number of rabbits increases by twice the number of rabbits the previous year.
➸ Answer a. [In the attachment]
➸ Answer b. Now, with the help of the table (attachment), we can take take the number of years as the variable x and the number of rabbits as the variable y. We also observe and get to know that the pattern that the variables follow is: when the years increase, the rabbits increases too. Hence, relationship between the variables that the table suggest is relationship of direct proportionality.
➸ Answer c. Assuming the number of rabbits in n years be y and the number of years be x. So, the equation formed will be : y = 20*2ⁿ.
➸ Answer d. The left side of the equation represents the number of rabbits. In the right side of the equation, 20 represents the initial number of rabbits. 2ⁿ present in the right side of the equation represents the pattern of increasing the number of rabbits by twice per year. Here, n represents year.
➸ Answer e. Using the equation, let find the number of rabbits in 20 years.
Number of rabbits = 20*2^(20).
We get 20*1,048,576.
The final number of rabbits = 20,971,520.
This may make sense in one way but we can also say this may not be possible as the rate of increase in rabbits may not be constant always. Also in these 20 years some rabbits may have died. Hence, in conclusion this doesn't make sense.
