PHYS 1407 – Conceptual Physics II

Exponential Growth

Leader: _________________________ Recorder: __________________________

Skeptic: _________________________ Encourager: ________________________

Materials

100 M&M’s

Styrofoam cup as a container

Paper plate

Laptop for graphs

Introduction

In a previous activity we explored the idea of exponential decay. An exponential decay occurs whenever the number of objects that decay is proportional to the total number of objects. Radioactivity provides a good example of a process that follows an exponential decay law.

Exponential growth is a very similar idea to exponential decay. An object grows exponentially if the number of new objects is proportional to the current number of objects. Many things grow exponentially at least for a short time. An important example is the number of fissioning nuclei in a chain reaction. Other examples of exponential growth include the world population of humans and the world consumption of fossil fuels. Exponential growth cannot usually be sustained indefinitely because the available resources to sustain that growth run out.

Procedure

1. Set-up

Use a pen or pencil to divide a clean paper plate into 10 compartments. Number the compartments 1 through 10.

Here is the procedure we will follow. Do not carry it out yet.

In the first compartment you will place 1 m&m, in the second 2, in the third 4, and in each subsequent compartment twice as many as in the previous.

P1) Do you have enough m&m’s to fill all 10 compartments? Explain your answer.

Place 1 m&m in the first compartment. Fill in the data in the table below. In the second compartment place 2 m&m’s, and then record the total number of m&m’s on the plate in the last column. Do not fill in the entire table at once. Fill in the rows as directed.

Q2) What will be the number of m&m’s that you add to the third compartment?

Q3) How does the number of m&m’s that you add to the third compartment compare to the total amount already on the plate.

Add the correct number of m&m’s to the third compartment.

Q4) By approximately what factor did the number of m&m’s change when you added the new amount.

Compartment Number/Step	Number of m&m’s placed in that compartment	Total number of m&m’s in all compartments
1
2
3
4
5
6
7
8
9
10

Q5) What will be the number of m&m’s that you add to the fourth compartment?

Q6) How does the number of m&m’s that you add to the fourth compartment compare to the total amount already on the plate.

Add the correct number of m&m’s to the fourth compartment.

Q7) By approximately what factor did the number of m&m’s change when you added the new amount.

Now complete the procedure adding twice the number of m&m’s in each successive compartment as in the previous one until you run out of m&m’s.

Q8) Did you have enough m&m’s to fill all 10 compartments?

Q9) Was your prediction P1) correct?

Complete the table above by calculating the correct number of m&m’s to add each time.

Q10) What would be the number of m&m’s required to finish the procedure?

D11) Use whatever software you like to make a properly labeled graph of Total number of m&m’s vs. step.

Q12) Describe the shape of the graph.

Print and attach the graph.

Just as we can characterize an exponential decay process by a half-life, we can characterize an exponential growth by a doubling time. The doubling time is the time required for the amount to double.

Q13) In terms of number of steps, what is the doubling time for this process?

Rule of 72

A rule of thumb for estimating the doubling time for an exponentially growing process is the rule of 72. If you know the rate of growth as a percentage, then the rule of 72 is that the doubling time in years is 72/growth rate. For example if a bank account earns 6%, the time for the account to double would be 72/6 = 12 or 12 years.

Q14) Austin is currently growing at a rate of 3% per year. If the growth rate remains the same, how many years will it take for Austin’s population to double?

Q15) If the city of Austin has a current population of 700,000 what will be its population in 24 years?

Exponential Growth and Finite Resources

If a resource is necessary to sustain exponential growth, then if that resource runs out, growth cannot continue.

Q16) How many compartments did you fill with the correct number of m&m’s?

Q17) Why didn’t you fill all of the compartments with the required number of m&m’s?

Q18) The current rate of growth of world use of petroleum is about 2% per year. If the rate continues, in how many years will the amount of petroleum used annually double?

Q19) If only half of the world’s reserves of petroleum have been used, assuming the same rate as in the previous question, in how many years will the remaining petroleum be used? Repeat if ¼ of the world’s petroleum reserves have been used.