For the scenario, identify key components, create an exponential function (y=a⋅bx), then answer the questions that follow.

Question

Suppose a rabbit population of 10 rabbits doubles every 6 months. 

1) a =

2) ​b = 

 3) Exponential Function:

4) How many rabbits will there be after 2 years (24 months)? 

5) How many rabbits will there be after 5 years (60 months)?

Answer 1

An exponential function is written

y = ab^x

So we need to find a and b.

Notice that at the start of the problem, we are zero months in. In that case, x=0. Substitute zero into the exponential function, recall that (almost anything)⁰ equals 1 and notice that

y = ab⁰

y = a(1)

y = a.

Next, recall that at the start of the problem, y=10. So we have found a = 10.

Now, if the population doubles every month, what do you expect to get when x=1? Take your answer to that, along with x=1 and solve for the base b.

Write back if you get stuck.

Answer 2

1) 10 (initial population)

2) 4 (since each year, the population will have quadrupled)

3) y = 10 * (4^x)

4) 160

5) 10240

Answer 3

10 rabbits double every 6 months

y=ab^x

y=10b^t where t=time in years

in 6 months there will be 20 rabbits

20=10b^[1/2] (1/2 year is 6 months)

2=b^[1/2]

2=√b

b=4

a=10

y=10(4)^t where t=time in years

y=10(4)^2

y=10*16

y=160 rabbits

y=10(4)^5

y=10*1024

y=10,240 rabbits

you can use t=months also

y=10b^[t/6]

20=10b^[t/6] where t=6 months

20=10b^[6/6]

2=b

a=10

y=10(2)^[t/6]