3.1+Reproducing+Rabbits

1. Open Software (Will usually have the brand name of the printer in its name) 2. Preview the Image you are about to Scan 3. Make the file smaller by cropping it 4. Make sure you have it set to a .png or .jpg format 5. Make sure the image has about 75 DPI
 * 12-7-08
 * NP
 * Big Idea: The rate at which things grow or decay can often be described mathematically.
 * Essential Question: How can I show a pattern of repeated division?
 * Notes from Class: How to Scan:

=Problem 3.1: Reproducing Rabbits=
 * = Time (years ||= Rabbit Population ||
 * = 0 ||= 100 ||
 * = 1 ||= 180 ||
 * = 2 ||= 325 ||
 * = 3 ||= 580 ||
 * = 4 ||= 1050 ||

A. The table above shows that the rabbit population grows exponentially. What is the growth factor for this rabbit population? The growth factor from one year to the next is the fraction: __population for year //n//__ population for year //n//-1

The growth factor for the rabbit population is approximately 1.8. I got this figure by dividing every rabbit population by the rabbit population in the row above it. For example, I divided 1050 by 580, and as an answer got 1.81. 580/325=1.78, 325/180=1.81, 180/100=1.8. Then I added all the quotients up, and divided them by 4, the number of quotients, to get an average. This average was 1.8


 * B. If this growth pattern had continued, how many rabbits would there have been in 10 years? After 25 years? After 50 years?**

If the growth factor of 1.8 continued, after

10 years there would be approximately 35,705 rabbits.

25 years there would be approximately 240,886,592 rabbits.

After 50 years there would be about 580,263,502,600,000 rabbits.


 * C. How many years would it take for the rabbit population to exceed 1 million?**

It would take about 16 years for the rabbit population to hit the 1 million benchmark. (1.8 to the power of 16 x 100 = 1214395)


 * D. Assume the growth pattern continued. Write an equasion you could use to predict the rabbit population, //P,// for any year, //n,// after the rabbits were first counted.**

The equasion for predicting the rabbit population at any time would be p= ( 1.8n ) x 100

P=Number of Rabbits 1.8=Growth Factor N=Year 100=Starting Value

=Follow Up=

a. With this growth factor, how many years would it have taken the population to grow from 1 million to 2 million?** It would take about 1.75 years for the population to reach 2 million. It would take about 4 years for the population to grow to 5 million. It would have taken between 5-6 years to grow to 10 million rabbits. It would take between 7-8 years for the population to reach 20 million rabbits.
 * [[image:lotsrabb.jpg align="left"]]1. Suppose that when the rabbit population reached 1 million, the government began to control progams that reduced the growth factor to 1.5
 * b. How many years would it have taken the population to grow from 1 million to 5 million?**
 * c. How many years would it have taken the population to grow from 1 million to 10 million?**
 * d. How many years would it have taken the popilation to grow from 1 million to 20 million?**

It would take about 4 years for the population to double. It would take about 1.75 years for the population to double. It would take a little more than a year for the population to double. As the growth factor increases, the less time it takes for the population to double.
 * 2. a. With a growth factor of 1.2, how many years would it have taken the population to double from any starting population?**
 * b. With a growth factor of 1.5**
 * c. With a growth factor of 1.8**
 * d. What observations can you make about the time it would take the population to double with growth factors of 1.2, 1.5, and 1.8?**

1.2T **) where //P// is the population in millions, and //t// is the number of years. a. What does this equasion assume about the population growth factor?** This equasion assumes the growth factor is equal to 1.2. This equasion assumes that the initial population was 15 rabbits.
 * 3. Suppose that during one time period, the rabbit population could be predicted by the equasion P=15(**
 * b. What does this equasion assume about the initial population?**

The percent is 50% The percent is 25% The percent is 10%
 * 4. You can think of a growth factor in terms of a percent change. For example, suppose the yearly growth factor for a rabbit population is 1.8**
 * **If the initial population is 100 rabbits, there will be 180 rabbits at the end of 1 year.**
 * **A change from 100 to 180 is an increase of 80 rabbits.**
 * **Since 80 rabbits is 80/100, or 80% of the original population, the rabbit population will increase by 80% yeach year**
 * Find the percent associated with each growth factor.**
 * a. 1.5**
 * b. 1.25**
 * c. 1.1**