Section 2. Working with Solutions: Modeling

Getting the solution formula is only the beginning of the story in most applications. Mathematical modeling requires solution manipulation. The exercises in this section are similar to the examples in the manual.

0. Pull down the Help menu, choose New Users/Full Tour, then click on the item four: 2-D Graphics.

Open a new Maple worksheet and do the following

1. Joe's savings account contains $12,000 dollars. He does not know the interest rate (compounded continuously) but after 60 days he checks and discovers that he has $12,130.

a. What is his annual interest rate?
b. How much money will be in the account one year from the day he has $12,130?

c. Plot the graph of P(t) including the points corresponding to 60 days and 60 days + 1 year.

Hint. First solve the exponential model initial value problem P' = r P, P(0) = 12000 with variable r and then substitute the data for t = 60/365 to determine the value of r. If you call the solution "soln" the substitution and r calculation can be made as follows

subs(P(t)=12130, t=60/365, soln);

r = evalf( solve(%,r) );

Then calculate P(1+60/365)) using

soln := subs(%,soln);  eval( soln, t=1+60/365);

The graph is the plot of rhs(soln) (and the two points).

2. Continuing 1. In Exercise 1 you discovered that Joe's savings account has an annual interest rate of 6.555 percent (approximately). Suppose that on the 60th day Joe also has a credit card debt of $560 at a 9% annual interest rate. Starting on that date he decides to pay off the credit card debt continuously from his savings account at the annual rate of $400 per year.

a. How many days later will the credit card debt be paid off?

b. At the time the credit card debt is paid off, how much money will Joe have in his savings account?

3.* The following problem is adapted from Ledder, Chapter 1, Section 1.

Suppose you borrow $12,000 to buy a car. The loan is to be paid in 60 equal monthly installments at an interest rate of 5% per year.

a. Assume the payments are actually made continuously at whatever rate is needed to pay off the loan in 60 months. Determine the continuous rate per month that would be required.

b. Compare the answer to part a to the answer if 60 equal monthly payments are made at a constant annual interest rate of 5% applied to the outstanding balance. In other words, the first payment, due one month after the loan is made, would be

12000*(exp(Float(5, -2)/12)-1)+P[1]

where P[1] is the amount that is put towards reducing the principal (the $12000) in the first month. The second payment is

(12000-P[1])*(exp(Float(5, -2)/12)-1)+P[2]

where P[2] is the amount put towards reducing the principal in the second month. The nth payment is

(12000-(sum(P[k], k = 1 .. n-1)))*(exp(Float(5, -2)/12)-1)+P[n] .

Hint for b. The second payment is supposed to equal to the first. Therefore, the following equation must be satisfied.

> eqn := 12000*(exp(0.05/12)-1)+P[1] =
      (12000 - P[1])*(exp(0.05/12)-1) + P[2];

eqn := 50.10431+P[1] = 50.10430800-0.4175359e-2*P[1]+P[2]

This determines P[2] in terms of P[1].

> P[2] := solve(eqn,P[2]);

P[2] := 0.2000000000e-5+1.004175359*P[1]

Create a for..do loop that calculates P[3], P[4], ... , P[60] in terms of P[1], then find P[1] using the fact that

sum(P[k], k = 1 .. 60) = 12000

Solution. The answer to part a. is $226.04 per month. The answer to part b. is $226.51 per month or about $30 more over the five years of the loan. See the Solutions.

4. Continuing 3. Consult with a banker and/or the internet to determine the amount a bank would charge per month for a 5% loan of $12000 over 60 months. Maple has a package called finance. Consult item 15 on the New Users/Full Tour and see if any of those procedures could be used.

5. The following problem is adapted from Ledder, Chapter 1, Section 1.

There is a power failure in your house at 1:00 P.M on a winter afternoon and your heating system stops working. The temperature in your house is 68 degrees F when the power goes out. At 10:00 P.M. the temperature in the house is down to 57 degrees F. Assume that the outside temperature is 10 degrees F.

a. Estimate the temperature in your house at 7:00 A.M. the next morning. Should you worry about your water pipes freezing?

b. Suppose the power goes on at 8:00 A.M. providing heat flow into the house that would increase the temperature at the rate of 10 degrees per hour if there were no heat loss. At what time will the temperature in the house be 68 degrees F again assuming the outside temperature stays at 10 degrees F throughout the day?

c. What is the answer to the question in part b if the heating system provides heat sinusoidally with a period of 3 hours and a maximum temperature increase of 10 degrees:

H(t) = 10-10*cos(2*Pi*t/3)

6. Verify that there is only one value of k that satisfies the k_equation on page 47 of the manual.

Hint. Plot, on the same set of axes, the left side and the right side of k_equation.