# Elementary Differential Equations and Boundary Value Problems - Boyce W.E.

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> 5. A tank contains 100 gallons of water and 50 oz of salt. Water containing a salt concentration

of 4 (1 + 2 sin t) oz/gal flows into the tank at a rate of 2 gal/min, and the mixture in the tank flows out at the same rate.

(a) Find the amount of salt in the tank at any time.

(b) Plot the solution for a time period long enough so that you see the ultimate behavior of the graph.

(c) The long-time behavior of the solution is an oscillation about a certain constant level. What is this level? What is the amplitude of the oscillation?

6. Suppose that a sum S0 is invested at an annual rate of return r compounded continuously.

(a) Find the time T required for the original sum to double in value as a function of r.

(b) Determine T if r = 7%.

(c) Find the return rate that must be achieved if the initial investment is to double in 8 years.

7. A young person with no initial capital invests k dollars per year at an annual rate of return r. Assume that investments are made continuously and that the return is compounded continuously.

(a) Determine the sum S(t) accumulated at any time t.

(b) If r = 7.5%, determine k so that $1 million will be available for retirement in 40 years.

(c) If k = $2000/year, determine the return rate r that must be obtained to have $1 million available in 40 years.

> 8. Person A opens an IRA at age 25, contributes $2000/year for 10 years, but makes no addi-

tional contributions thereafter. Person B waits until age 35 to open an IRA and contributes $2000/year for 30 years. There is no initial investment in either case.

(a) Assuming a return rate of 8%, what is the balance in each IRA at age 65?

(b) For a constant, but unspecified, return rate r, determine the balance in each IRA at age 65 as a function of r.

(c) Plot the difference in the balances from part (b) for 0 < r < 0.10.

(d) Determine the return rate for which the two IRA’s have equal value at age 65.

9. A certain college graduate borrows $8000 to buy a car. The lender charges interest at an annual rate of 10%. Assuming that interest is compounded continuously and that the borrower makes payments continuously at a constant annual rate k, determine the payment rate k that is required to pay off the loan in 3 years. Also determine how much interest is paid during the 3-year period.

10. A home buyer can afford to spend no more than $800/month on mortgage payments. Suppose that the interest rate is 9% and that the term of the mortgage is 20 years. Assume that interest is compounded continuously and that payments are also made continuously.

(a) Determine the maximum amount that this buyer can afford to borrow.

(b) Determine the total interest paid during the term of the mortgage.

11. How are the answers to Problem 10 changed if the term of the mortgage is 30 years?

2.3 Modeling with First Order Equations

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> 12. A recent college graduate borrows $100,000 at an interest rate of 9% to purchase a con-

dominium. Anticipating steady salary increases, the buyer expects to make payments at a monthly rate of 800(1 + t /120), where t is the number of months since the loan was made.

(a) Assuming that this payment schedule can be maintained, when will the loan be fully paid?

(b) Assuming the same payment schedule, how large a loan could be paid off in exactly 20 years?

13. A retired person has a sum S(t) invested so as to draw interest at an annual rate r compounded continuously. Withdrawals for living expenses are made at a rate of k dollars/year; assume that the withdrawals are made continuously.

(a) If the initial value of the investment is S0, determine S(t) at any time.

(b) Assuming that S0 and r are fixed, determine the withdrawal rate k0 at which S(t) will remain constant.

(c) If k exceeds the value k0 found in part (b), then S(t) will decrease and ultimately become zero. Find the time T at which S(t) = 0.

(d) Determine T if r = 8% and k = 2k0.

(e) Suppose that a person retiring with capital S0 wishes to withdraw funds at an annual rate k for not more than T years. Determine the maximum possible rate of withdrawal.

(f) How large an initial investment is required to permit an annual withdrawal of $12,000 for 20 years, assuming an interest rate of 8%?

14. Radiocarbon Dating. An important tool in archeological research is radiocarbon dating. This is a means of determining the age of certain wood and plant remains, hence of animal or human bones or artifacts found buried at the same levels. The procedure was developed by the American chemist Willard Libby (1908-1980) in the early 1950s and resulted in his winning the Nobel prize for chemistry in 1960. Radiocarbon dating is based on the fact that some wood or plant remains contain residual amounts of carbon-14, a radioactive isotope of carbon. This isotope is accumulated during the lifetime of the plant and begins to decay at its death. Since the half-life of carbon-14 is long (approximately 5730 years1), measurable amounts of carbon-14 remain after many thousands of years. Libby showed that if even a tiny fraction of the original amount of carbon-14 is still present, then by appropriate laboratory measurements the proportion of the original amount of carbon-14 that remains can be accurately determined. In other words, if Q(t) is the amount of carbon-14 at time t and Q0 is the original amount, then the ratio Q(t)/Q0 can be determined, at least if this quantity is not too small. Present measurement techniques permit the use of this method for time periods up to about 50,000 years, after which the amount of carbon-14 remaining is only about 0.00236 of the original amount.

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