## MATH105: Exploring Exponential Functions

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The purpose of this exercise is to generate data on exponential growth and decay and to find equations that describe that data. Please work in pairs to collect your data; you may combine into larger groups to work on the later portion of the worksheet.

Population Growth (20 pts)

In this exercise you will observe the growth of a population of M&M's. These M&M's live in a perfect habitat: they never age or die, and in every generation each M&M has a 50% chance of producing a child M&M. Please do not eat the M&M's -- you will need them for the second activity.

 Generation 0 1 2 3 4 5 6 7 No. of M&M's 4

You start with a population of 4 M&M's. Place 4 M&M's in a cup and shake the cup gently, then carefully pour the M&M's onto a paper plate.

Count the number of M&M's on the plate that have an M showing. Add one M&M to the cup for every M you see. These M&M's simulate the children of the M-side-up M&M's. Return the 'adult' M&M's to the cup and record the total number of M&M's now in the cup.

Repeat the process described above until you run out of M&M's. At that point, record the number of M&M's you would have needed to have in the cup to continue.

Plot your population sizes on the chart below.

Endangered Species (20 pts)

Not all M&M's are fortunate enough to live in a perfect habitat. You will now study the population of a community of M&M's whose size is shrinking rapidly!

 Generation 0 1 2 3 4 5 6 7 8 9 No. of M&M's

Count your M&M's and place them in the cup. Record the initial number of M&M's as the population of Generation 0 in the table above.

Shake the cup and pour the M&M's onto the paper plate. M&M's that have an M showing survive -- count them, record the new population size, and put them back into the cup. The M&M's with no M showing are not so lucky. Dispose of their little chocolatey corpses as you see fit.

Repeat the process until your M&M's are extinct. Plot your population data in the chart below.

Making A Mathematical Model (60 pts)

If M&M's are allowed to reproduce unhindered in an ideal environment, how long will it take for their growing numbers to become a chocolatey menace to society? How long will it take brave M&M extermination teams to eliminate this threat to our waistlines?

Find mathematical models of the populations you studied and record them on the next page. You may use algebra (assume there's a 50% or 1 in 2 chance of an M&M landing M up), a curve fitting tool on your calculator or Graphmatica, or a verbal description.

1. Model of endangered M&M population:

• If you have p=60 M&M's in your cup, about how many M&M's will land M-side up when you pour them out?

• If have 60 M&M's in generation p and only M-side-up M&M's survive to generation q, approximately what multiple of p survives to generation q?
•

p x ______ = q
• In a hostile environment, M&M population is (approximately) multiplied by ______ in each generation.

• Use the multiple you found above to predict the number of M&M's in generation 0, 1, 2 and 3 if there are 60 M&M's in generation 0.
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• Write an equation describing the M&M population in generation n using your starting population.

• What population does your equation predict for generation 4? Does this match your observations?

2. M&M population in an ideal environment:

• If you have p=60 M&M's in your cup, about how many M&M's will land M-side up when you pour them out?

• If all of these "reproduce", about how many M&M's will you have in your cup in the next generation?

q=______

• What multiple of p is q?

p x ______ = q

• In an ideal environment, M&M population is (approximately) multiplied by ______ in each generation.

• Use the multiple you found above to predict the number of M&M's in generation 0, 1, 2 and 3 if there are 4 M&M's in generation 0.
•

• Write an equation describing the M&M population in generation n if the population is 4 in generation 0.

• What population does your equation predict for generation 4? (Plug 4 into your equation.) Does this match your observations?
•

3. Professor Burgiel left a packet of 25 M&M's on her desk. Suppose Prof. Burgiel's office is an ideal environment for an M&M population. How many generations of reproduction will it take before she has over 50 M&M's on her desk? Over 100? Over 200? (Please show your work!)

Bonus Suppose each M&M covers approximately 1 cm² of area and that the area of Professor Burgiel's office is 135,000 cm² = 13.5 m². How many generations of M&M reproduction will it take to completely cover her office?