In the year 2005, the population of rabbits was 6900, and by 2012 the population had grown to 13500

Question

The population of rabbits on an island is growing exponentially. In the year 2005, the population of rabbits was 6900, and by 2012 the population had grown to 13500. Predict the population of rabbits in the year 2015, to the nearest whole number.

Answer 1

Ok so this is a case where we need to create the equation for the exponential function based on only being given 2 known datapoints.

We are given population data for 2 years, so first, always make time the independent variable x, and the other variable y:

In (x,y) form:

(year, # of rabbits)

1: (2005, 6900)

2: (2012, 13500)

We find the equation for exponential functions with the following approach:

First, build 2 equations with unknown variables a & b:

y₁=a*b^x1

y₂=a*b^x2

Plugging in our given values for x and y for the 2 given datapoints:

6900 = a*b²⁰⁰⁵

13500 = a*b²⁰¹²

Next we need to solve for the variables a & b for our final equation.

The final equation will be in the format: y=a*b^x

Looking at the 2 equations above, we need to combine them using substitution or another method. Substitution is not the easiest way for this problem. Since both equations contain variable a, let's divide the first equation by the second equation, so the variable a will cancel out:

6900 = a*b²⁰⁰⁵

(divided by) = 6900/13500 = b²⁰⁰⁵ / b²⁰¹²

13500 = a*b²⁰¹²

Next, recall the division property of exponents:

x^y/x^z=x^y-z

Apply this to the [b²⁰⁰⁵ / b²⁰¹²]:

Subtract 2005-2012= -7

So:

6900/13500 = b^-7

Now use properties of exponents to solve for b:

6900/13500 = b^-7

13500/6900 = b⁷

Use calculator for 7th root of 13500/6900 = 1.1

b = 1.1

Now, plug this value of b into the final equation for the rabbits, but we still need to solve for a:

y=a*1.1^x

To determine a, choose either of the 2 given datapoints and plug in:

6900=a*1.1²⁰⁰⁵

Now use a calculator to evaluate 1.1²⁰⁰⁵, then divide 6900 by that number to solve for a:

1.1²⁰⁰⁵ = 9.825 x10⁸²

6900=a*9.825 x10⁸²

a=6900/(9.825 x10⁸²)

a=7x10^-80

Now we have our final equation for the rabbits:

y=(7x10^-80)*1.1^x

Notice if you plug in 2005 or 2012 into this equation as x, you will get 6900 and 13500 respectively for y.

The error is due to rounding.

This equation is modeling the population of rabbits correctly.

Finally, we can use this equation to predict the population of rabbits in 2015.

Plug in 2015 for x:

y=(7x10^-80)*1.1²⁰¹⁵

Use a calculator to solve for y:

y = 17838.512

Round to the nearest whole number:

Population of rabbits in 2015: 17839

If I run through this problem again without rounding any answers, I get these numbers:

b = 1.100628282

a = 2.23516324x10^-80

So,

y=(2.23516324x10^-80)*1.100628282^x

Plug in 2015 for x:

y=(2.23516324x10^-80)*1.100628282²⁰¹⁵

Exact answer:

y = 17999.307

Round to the nearest whole number:

Population of rabbits in 2015: 17999

Hope this helps! :)

Answer 2

The general equation for exponential change in the context of population dynamics is:

P(t)=Ae^(kt)

Where 'P(t)' represents the population at time 't', 'A' is the population at time t=0, 'e' is Euler's number 2.718..., and 'k' is a constant parameter which determines the rate of change of the population with respect to time.

In this example, if we let t=0 represent the year 2005, then A=6900. In order to find a general equation for P(t), we need to also determine 'k'. 2012 is 7 years later than 2005, so using the second data point (7,13500), we have:

P(7)=13500=6900e^(7k)

Solving for 'k' yields:

k=(1/7)*ln(13500/6900)

Now, we use this value of the parameter to predict the population in 2015:

P(10)=6900e^[(1/7)*ln(13500/6900)*10]=17999.3≈18000

Answer 3

Hi Gabriella!

Rabbit populations are often held up as an example of exponential growth. Let R equal the rabbit population, let t equal time in years, and let b be some base of your exponential function. We then write:

R = R(0)*b^t

where R(0) is the initial number of rabbits, and b^t is the base raised to the power of t. The base b is a number that is above one for exponential growth, and less than one for exponential decay.

At first, in 2005, your rabbit population was 6900, and that's R(0). We cannot find b from this, because b^0 is one. You are given a second point, in 2012, and that is

R(2012-2005) = R(7) = R(0) * b^7

You then just divide R(7) by R(0), and find the seventh root of the ratio. Your calculator may have a caret ("^") or an "x^y" or even an "x^(1/y)" key you can use to do this. This will give you the value of b.

Now you are almost done; R(2015-2005) = R(10) is simply R(0)*b^10.

Cheers,

Joe F.