1 Group Project Problems

In your group, select one of the following problems you want to work on together. Work together to find a solution (or solutions) to the given problem(s) and/or related questions you find interesting. Then, as a group, write a 1-2 page write-up presenting:

  • the problem or question you worked on,
  • a solution and the tools and reasoning you used to arrive at a solution, and
  • the significance of the result and how it can contribute toward better decision-making.

Make sure you answer all the questions in your problem and edit your write-up to make sure it is readable with no grammar or spelling errors.

Each group will also make a short video presentation of their work, so keep in mind, that your work will be made public for other students to view and study.

1.1 Fish Population Management

Recursive equations can be very handy for modeling complicated situations for which explicit equations would be hard to interpret. As an example, consider a lake in which 2000 fish currently reside. The fish population seems to be growing by 10% each year, but every year about 100 fish are harvested from the lake by people fishing.

  1. Write a recursive equation for the number of fish in the lake after n years.

\(P_{n+1}= P_n*1.1-100\)

  1. Calculate the population after the first 10 years. What does the population appear to be doing in the long run?
      n   Logistic  Recursive 1  
      1   2000      2100  
      2   2100      2210  
      3   2200      2331  
      4   2299      2464.1  
      5   2397      2610.51   
      6   2493      2771.561  
      7   2587      2948.7171   
      8   2678      3143.58881  
      9   2766      3357.947691  
      12  2852      3593.74246  
      11  2934      3853.116706
  1. Do you think this model will be accurate in the long run?

Answers vary.

  1. What is the number of fish that could be harvested each year without causing the fish population to change in the long run?

Answers vary.

  1. Officials determine that 4000 fish is the ideal population for this lake in terms of the overall health of the ecosystem surrounding it. What recommendations would you have for harvesting fish from this lake?

400

1.2 Forensic Science

A hotter object in a colder room will decrease in temperature exponentially, approaching the room temperature according to the formula \[T_n = a(1-r)^n+T_r,\] where:

  • \(T_n\) is the temperature of the object after \(n\) minutes,
  • \(r\) is the rate at which temperature is changing,
  • \(a\) is a constant, and
  • \(T_r\) is the temperature of the room.

Forensic investigators use this formula to predict the time of death of a homicide victim. Suppose that when a victim was discovered (\(n = 0\)), the body was 85 degrees F. After 20 minutes, the temperature of the body was measured again to be 80 degrees. The temperature of the room was 70 degrees F.

  1. Use the given information above to find a formula for the temperature of the body. Hint: First, use the information \(T_n=85\) at \(n=0\) to find \(a\). (Remember anything to the zero power is 1). Then, use the information \(T_n = 80\) at \(n = 20\) to find \(r\), possibly using the Desmos graphing calculator or an online equation solver (here is one). After solving for \(a\) and \(r\), you have a formula for \(T_n\) in terms of n!
a = 15
r = 1-(10/15)^(1/20)
r
## [1] 0.02006913
  1. If the body started at 98.6 degrees F, when did the victim die?

Solve 98.6=15(1-0.02006913)^n+70

n = log((98.6-70)/15)/log(1-r)
n 
## [1] -31.8329

Around 31.83 minutes before body was found.

1.3 Logistic Growth

Read the section on Logistic Growth and/or find some videos that explain logistic growth (e.g. here is one example). Use this information to explain what logistic growth is when it occurs, and what formula(s) are used to model it. Then use this understanding to answer the following questions.

  1. One hundred trout are seeded into a lake. Absent any constraints, the trout population will grow by 70% a year. The lake can sustain a maximum of 2000 trout. Using the logistic growth model,
    1. Write a recursive formula for the number of trout
    2. Use your recursive formula to calculate the number of trout after 1, 2, 3, 4, and 5 years.
  2. Now model the same fish population growing 70% per year without constraint, and calculate the number of trout after 1, 2, 3, 4, and 5 years.
  3. Compare your answers to 1. and 2. above using a table or graph. Discuss the reason for any noted differences.

1.4 Should you Refinance?

Your original mortgage

Suppose that 10 years ago you bought a home for $130,000, paying 10% as a down payment, and financing the rest at 6% interest for 30 years.

1a. On your existing mortgage (the one you got 10 years ago), how much money did you pay as your down payment?

130000*0.1
## [1] 13000

1b How much money was your existing mortgage (loan) for?

P0 = 130000*0.9
P0
## [1] 117000

1c. What is your current monthly payment on your existing mortgage?

d = round( P0/( (1-(1+0.06/12)^(-12*30) ) / (0.06/12) ), 2)
d
## [1] 701.47

1d. How much will you pay in monthly payments over the life of this existing loan?

d*12*30
## [1] 252529.2

1e. How much total interest will you pay over the life of this existing loan?

d*12*30-P0
## [1] 135529.2

Where are you now?

Now, 10 years after you first took out the loan, you check your loan balance. Only part of your payments have been going to pay down the loan; the rest has been going toward interest. You see that you still have $102,637 left to pay on your loan. Your house is now valued at $180,000.

At this point:

2a. How much of the original loan have you paid off?
(i.e., how much have you reduced the loan balance by? Keep in mind that interest is charged each month - it’s not part of the loan balance.)

paid_off = 117000-102637
paid_off
## [1] 14363

2b. How much money have you paid to the loan company so far (over the last 10 years)

paid_total = 10*12*d
paid_total
## [1] 84176.4

2c. How much interest have you paid so far (over the last 10 years)?

paid_interest = paid_total-paid_off
paid_interest
## [1] 69813.4

2d.How much equity do you have in your home (equity is value minus remaining debt)?

equity = 180000-102327
equity
## [1] 77673

Refinancing

Since interest rates have dropped, you consider refinancing your mortgage at a lower 4.5% rate. To refinance, you would pay a refinancing fee of 2% of the total mortgage, which is added to your new mortgage amount. You also notice that if you refinance, you are going to be making payments on your home for another 30 years. Note that these 30 years of payments are in addition to the 10 years you’ve already been paying, making 40 years of total payments.

If you took out the new 30-year mortgage at 4.5%:

3a. How much would your new mortgage be?

P_o = 102637*1.02
P_o
## [1] 104689.7

3b. What would your new monthly payments be?

d = round( P_o/( (1-(1+0.045/12)^(-12*30) ) / (0.045/12) ), 2)
d
## [1] 530.45

3c. How much will you pay in monthly payments over the life of the new loan?

d*12*30
## [1] 190962

3d. How much interest would you pay over the life of the new loan?

new_interest = d*12*30-P_o
new_interest
## [1] 86272.26

3d. How much less will you pay each month because of the lower monthly payment?

701.47-530.45
## [1] 171.02

3e. How much total interest will you be paying on house payments? Consider the interest you paid over the first 10 years of your original loan, as well as interest on your refinanced loan.

paid_interest+new_interest
## [1] 156085.7

Conclusion

Discussion of your answers to the above questions. Conclude with a discussion of the question: Should you refinance? (There isn’t a correct answer to this. Give your opinion and justification.) Some questions to consider would be:

Some questions to consider would be:

  • how much total money are you paying in each case?
#Original Loan
252529.2
## [1] 252529.2
#Original Loan + Refinance
84176.4 + 190962
## [1] 275138.4
#difference = total more paid to refinance
84176.4 + 190962 - 252529.2
## [1] 22609.2
  • how much interest are you paying in each case?
#Original Loan interest
135529.2
## [1] 135529.2
#Original Loan interest + Refinance interest
69813.4 + 86272.26
## [1] 156085.7
#difference
(69813.4 + 86272.26) - 135529.2
## [1] 20556.46
#extra interest + finance fee = total more paid to refinance
(69813.4 + 86272.26) - 135529.2 + 102637*0.02
## [1] 22609.2

1.5 Maxing out your Credit

Suppose that as a Freshman you apply for the Chase Unlimited Freedom credit card and receive your credit card with a $2000 credit limit. This credit card has a 0% annual percentage rate (APR) for the first 15 months, meaning you don’t pay ANY interest during this time. After that, if you do not pay your bill, you owe 30% APR, which is accrued monthly.

After these 15 months, you have maxed out your credit and have a $2000 bill due the next month. You think perhaps you should wait to pay your credit card off until you graduate and get a job to pay for it. (In this case, where no payments are made, there are often additional fees owed; however, we will not consider this here in this example.)

  1. Suppose after 15 months, you decide not to make your minimum payment on your credit card:
    1. In this case, you will owe a 30% annual interest rate. How much of the 30% annual interest is owed after one month?
    2. How much total interest do you owe after the first month?
    3. How much is your credit card balance after the first month?
  2. Suppose you cannot pay your credit card after the first two months.
    1. How much interest do you owe after the second month?
    2. How much do you owe on your credit card after the second month?
  3. Suppose you cannot pay your credit card after the first three months.
    1. How much interest do you owe after the third month?
    2. How much do you owe on your credit card after the third month?
  4. Use Google Sheets or Microsoft Excel to repeat the calculations from steps 2-3, for up to 36 months!
    1. Follow the steps described in this video or together in class.
    2. Based on your calculations, how much do you owe after 36 months?
    3. How much interest have you accrued?
  5. In this step, we’ll use the compound interest formula to calculate how much you would owe after 36 months.
    1. If you have not yet, familiarize yourself with the compound interest formula. Identify what each variable means.
    2. Use the compound interest formula to calculate how much you owe after 36 months.
    3. Compare with your answer in 4b.
  6. Suppose after 15 months, you decide to get a part-time job to pay $100/month on your credit card. Let’s do an experiment and compute how long it would take to pay off your credit card that way.
    1. Modify the formula in cell B3 to subtract 100 from each month’s balance.
    2. Then copy the formula in cell B3 down as you did in step 4b(iii).
    3. How long does it take to pay off your bill at this rate?
    4. After the last month in which you would pay your credit card off, how much total money would you have paid? (Make sure you include the total of all the payments you have made plus the last month’s interest.)
    5. At this point, how much total interest have you paid? Compare with your answer from 4c.
  7. (Optional) Suppose that, from the very beginning of college, you decided to get a part-time job and paid $100/month on your credit card. Let’s find out how much you would pay that way.
    1. Suppose again, you spend $2000 on your credit card during the first 15 months, but pay $100/month b. to pay it off. How much do you owe on your credit card after the first 15 months?
    2. If you continue to pay $100/month on your credit card after the first 15 months, how long would it take to pay it off?
    3. After the last month in which you would pay your credit card off, how much total money would you have paid? (Again, make sure you include the total of all the payments you have made plus the last month’s interest.) By the time you pay your credit card off, how much interest have you paid?

Based on this, discuss with your group/class how this informs your approach to credit cards and loans. Write a 2-paragraph summarizing: The tools used for finding your results. Results and their significance. How can these results contribute to better decision-making?

You are shopping for a car for $20,000 and have enough money in your account to pay upfront, but then your salesman says he can get you a low-interest loan, 4% for 3 years. He argues that by investing your $20,000 at a higher interest rate, you will come out ahead. Is he right?

You are smart and know to expect hidden fees. You ask the salesman, “What are the fees for the loan?”

After rummaging around in the back room, the salesman comes back and says, “You’d have a 3% fee,” which would be added to the total loan amount.

So, you have two options:

  • Option 1: Take the loan and invest the $20,000 elsewhere.
  • Option 2: Do not take the loan, but invest what you would be paying every month.

Question 0: Based on your understanding of the problem, in which option do you think you would end up with more money? Why?

Now, to determine in which case you will come out ahead, you decide to compute your 3-year gain in each case.

To determine in which case you will come out ahead, you decide to compute your 3-year gain in each case.

1.5.1 Option 1 - Take the Loan

If you take the loan:

1a) How much is the total loan with 3% lender fees included?

P_o = 20000*1.03
P_o
## [1] 20600

1b) What would be your monthly payment for this loan?

r = 0.04 N = 3 k = 12

$608.19

r = 0.04
N = 3
k = 12
RHS = (1-(1+r/k)^(-N*k))/(r/k)

# monthy payments
d = round(P_o/RHS,2)
d
## [1] 608.19

1c) How much would you pay over the course of this loan?

d*12*3
## [1] 21894.84

If you take the loan and follow the salesman’s advice, you can invest the $20,000 from your bank account straight into index funds.

1d) Suppose you invest the $20,000 and earn 5% annual interest, compounded monthly. How much would this investment be worth after 3 years?

20000*(1+0.05/12)^(12*3)
## [1] 23229.44

1.5.2 Option 2 - Do not take the loan

If you do not take the loan, you would have no monthly payment, so you could invest the same amount of money as your monthly payment (from 1b) each month.

2a) If you invest the money you would be paying against your car payment (from 1b) into index funds each month (earning 5% annual interest, compounded monthly), how much this fund be worth after 3 years?

d
## [1] 608.19
r = 0.05
N = 3
k = 12
d*((1+r/k)^(N*k)-1)/(r/k)
## [1] 23569.39

1.5.3 Your decision

  1. Was the salesman right? Which option is better? By how much would you come out ahead? By how much would you come out ahead? What would you choose and why?

1.5.3.1 Option A

$23229.44

1.5.3.2 Option B

$23569.39

23569.39 - 23229.44
## [1] 339.95

Net if you take the loan:

23339.44 - 21894.84
## [1] 1444.6

Net if you do not take the loan but:

23569.39 - 21894.84
## [1] 1674.55
1674.55 -1444.6
## [1] 229.95

You would come out $229.95 ahead if you took the car loan. If I wanted to build credit, I’d take the loan.

1.5.4 Feedback

Scenario 1 - Take the loan - Your work on this is flawless!

Scenario 2 - Do not take the loan - Your work on the formula looks great, but because of rounding, I got an answer one cent different than yours: $23569.39

Conclusion - Check your reasoning. For scenario 1, you paid $21,894.84 in payments over 4 years, but ended up with $23569.39 in your bank. For scenario 2, you paid the same amount in monthly investments, and ended up with $23569.39 in your bank. In this case

1.6 Payday loans

Payday loans are short-term loans that you take out against future paychecks: The company advances money against a future paycheck. First, read more here to see how payday loans work.

Check ’n Go is one example of a payday loan loan company. Here are example loan rates for getting a Check ’n Go payday loan online. Use this rate (360% APR) with biweekly payment schedule for this problem.

Notice that if you are paying your loan after 14 days (2 weeks), this is the period. Since there are 52 weeks in a year, for this loan, there are 52/2 = 26 periods per year.

  1. Suppose you decide to borrow $500 and that we will pay back the loan in 14 days. Determine the total amount that you would need to pay back and the effective loan rate.

  2. If you cannot pay back the loan after 14 days, you will need to get an extension for another 14 days, paying an additional 14 days of interest on the original amount you owe in (a). Determine the total amount you will be paying for the now 28-day loan.

  3. Suppose you wait a year to pay off this loan. How much are you paying?

In your write-up, include a discussion of the above question. Make sure you explain the general way in which payday loans work and your conclusions about whether or not (or when and when not) payday loans are a smart thing to do.