Week #15 - Word Problems & Differential Equations Section 8.6 From Calculus, Single Variable by Hughes-Hallett, Gleason, McCallum et. al. Copyright 5 by John Wiley & Sons, Inc. This material is used by permission of John Wiley & Sons, Inc. SUGGESTED PROBLEMS 1. Find the future value of an income stream of $1 per year, deposited into an account paying 8% interest, compounded continuously, over a 1-year period. We are adding money at a constant rate of $1 per year. The amount deposited in the small time interval t at time t will be $1 t. This money will grow at 8% interest for the remaing 1 t years, so its future value will be ($1 t) e.8 (1 t) Adding up all the deposits from t = to t = 1, and letting t dt, we get Future value = 1 1e.8(1 t) dt.8(1 t) 1 = 1e 1.8 = 1 ( e + e.8(1)).8 = $15, 319.6 This is noticeably more than the actuall $1, deposited, and seems reasonable for the amount of interest that should have been earned. 3. Find the present and future values of an income stream of $ a year, for a period of 5 years, if the continuous interest rate is 8%. Similarly to #3, the future value will be given by Future value = 5 e.8(1 t) dt.8(5 t) 1 = e 5.8 = ( e + e.8(5)).8 = $1, 95.6 Note that this value is less than for the 1 year case in #1, even though the total deposit is still $1,. This is because there is only five years for the interest to accumulate. 1
The present value can be computed more easily using the simple relationship on page 411, that Future Value = e rm Present value where r is the interest rate and M is the number of years. Future Value Present value = e rm = $1,95.6 e.8 5 $8,4. 9. A business associate who owes you $3 offers to pay you $8 now, or else pay you three yearly installments of $1 each, with the first installment paid now. If you use only financial reasons to make your decision, which option should you choose? Justify your answer, assuming a 6% market interest rate, compounded continuously. You should choose whichever payment schedule maximizes it either present or current value. Since one of the options is a lump-sum payment of $,8 right now (and so it s present value is $,8), it would be easier to compare present values. When we consider the three payments of $1,, we need to reduce each of their value from the future to their present value. One payment is immediate, so has a present value of $1,. The next payment is a year from now, so needs to be discounted by e.6 1. The last payment will be given in two years, so needs to be discounted by e.6. The net present value of all three payments then is $1, + $1, e.6 1 + $1, e.6 $88.68 From this analysis, you are marginally better off taking the payment in $1, installments, although the difference is relatively slight $8 on the $3, amount. You would still be taking a loss relative to what you are owed. QUIZ PREPARATION PROBLEMS 7. (a) A bank account earns 1% interest compounded continuously. At what (constant, continuous) rate must a parent deposit money into such an account in order to save $1, in 1 years for a child s college expenses? (b) If the parent decides instead to deposit a lump sum now in order to attain the goal of $1, in 1 years, how much must be deposited now? (a) The future value of a continuous income stream is given by the fomrula on page 411 of the text. In this case, the income stream is constant, so P(t) = R, and we are looking for R such that the future value equals $1,.
Future value = 1 1, = Re 1, = R.1 Re.1(1 t) dt.1(1 t) 1 1.1 [ e + e.1(1)] R = 1, e 1 $5,819.77 per year e The parents would need to deposit approximately $5,819.77 per year to achieve their goal. (b) A single lump sum of P, over 1 years, would grow to P e.1(1). To find an initial deposit that would grow to 1, in ten years, we solve P e 1 = 1, P = 1, e 1 $36,787.94 They would need to deposit $36,787.94 today to have $1, in ten years. 13. An oil company discovered an oil reserve of 1 million barrels. For time t >, in years, the company s extraction plan is a linear declining function of time as follows: q(t) = a bt where q(t) is the rate of extraction of oil in millions of barrels per year at time t and b =.1 and a = 1. (a) How long does it take to exhaust the entire reserve? (b) The oil price is a constant $ per barrel, the extraction cost per barrel is a constant $1, and the market interest rate is 1% per year, compounded continuously. What is the present value of the company s profit? (a) If the extraction rate is q(t) = 1.1t millions of barrels per year, the total amount extracted in N years will be N 1.1t dt We want that amount to equal 1 ( millions of barrels is already in the units), 3
and solve for N: 1 = N 1.1t dt 1 = 1t.1t 1 = 1N.1N Set up as quadratic: =.5N 1N + 1 Use quadratic formula: N = 1 ± 1 4(.5)(1) (.5) N = 1.6, 189.4 years Only the 1.6 year estimate makes sense, as the rate would be negative at 189 years. If the estimate of 1 million barrels is correct, it should take 1.6 years to exhaust the well. (b) If oil sale price is $ per barrel, and it costs $1 to extract, then every barrel of oil will produce a profit of $ - $1 = $1. This means that we can translate the rate of oil into the rate of money flowing in to an account, with P(t) = rate of deposit = $1 q(t) = $1(1.1)t = $(1 t) millions of dollars per year This income stream will be flowing as long as the well is pumping, which we found would be for 1.6 years in part (a). Using the present value formula on page 41 of the text, we can compute the present value of all the oil that will be produced over that time. Present value = = Split into two integrals: = 1.6 1.6 1.6 P(t)e rt dt (1 t)e rt dt 1e rt dt 1.6 te rt dt The first integral is 1e rt /r. The second integral requires integration by parts: selecting: u = t dv = e rt dt so du = dt v = e rt /r Using the integration by parts formula, 4
te rt dt = te rt /r + 1 r e rt dt = te rt /r 1 r ( e rt so the Present value = 1 e rt t r r e rt 1 ) 1.6 r e rt ( 1 = e rt r + t 1 ) r ( ) 1 with r = 1% =.1, = e 1.6 1.6 1.1.1 ( 1 e.1 1 ).1 $64.9 million The present value is roughly $65 million dollars. This seems reasonable, since the total value of the oil is $1, million dollars ($1 for each of the 1 million barrels). However, the oil will only be pumped out later, so the present value is slightly discounted. 5