171S4.5.notebook
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MAT 171
4.5 Applications from Business, Finance, and Science
A. Simple and Compound Interest
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B. Interest Compounded Continuously
C. Applications Involving Annuities and Amortization
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D. Applications Involving Exponential Growth and Decay Growth Rate
Closely related to interest compounded continuously are applications of exponential growth and exponential decay. If Q (quantity) and t (time) are variables, then Q grows exponentially as a function of t if Q(t) = Q0e rt for positive constants Q0 and r.
Decay Rate
If Q(t) = Q0e ­rt, then we say Q decreases or decays exponentially over time. The con­
stant r determines how rapidly a quantity grows or decays and is known as the growth rate or decay rate constant.
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Exploring Compound Interest
How long would it take $1000 investment to double at 7% compounded monthly? If it doubles, the amount would be $2000 (or 2P) at the end of t years. Use the formula
The TI screens below show (1) the values for P, r, and n are stored; (2) Y1 as the amount formula and Y2 as the amount after doubling; (3) the window settings for this problem; and (4) the graph showing the intersection of Y1 and Y2. So, it would take almost 10 years (X = 9.93) for any amount of investment to double at a rate of 7% compounded monthly.
Screen 1
Screen 2
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406/8. For simple interest accounts, the interest earned or due depends on the principal p, interest rate r, and the time t in years according to the formula I = prt.
Find r given I = $1928.75, p = $8500, and 3.75 yr.
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406/10. For simple interest accounts, the interest earned or due depends on the principal p, interest rate r, and the time t in years according to the formula I = prt.
Angela has $ 750 in a passbook savings account that pays 2.5% simple interest. How long will it take the account balance to hit the $ 1000 mark at this rate of interest, if she makes no further deposits? ( Hint: How much interest will be paid?)
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406/12. For simple interest accounts, the amount A accumulated or due depends on the principal p, interest rate r, and the time t in years according to the formula A = p(1+rt).
Find r given A = $15,800, p = $10,000, and t = 3.75 yr.
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406/14. For simple interest accounts, the amount A accumulated or due depends on the principal p, interest rate r, and the time t in years according to the formula A = p(1+rt).
Healthy U sells nutritional supplements and borrows $ 50,000 to expand their product line. When the note is due 3 yr later, they repay the lender $ 62,500. If it was a simple interest note, what was the annual interest rate?
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406/18. For accounts where interest is compounded annually, the amount A accumulated or due depends on the principal p, interest rate r, and the time t in years according to the formula A = p
(1+r)t.
Find p given A = $30,146, r = 5.3%, and t = 7 yr.
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406/20. For accounts where interest is compounded annually, the amount A accumulated or due depends on the principal p, interest rate r, and the time t in years according to the formula A = p(1+r)t.
What interest rate will ensure a $ 747.26 deposit will be worth $ 1000 in 5 yr?
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406/24. For compound interest accounts, the amount A accumulated or due depends on the principal p, interest rate r, number of compoundings per year n, and the time t in years according to the formula A = p(1+r / n)nt.
Find r given A = $95,375, p = $65,750, and t = 15 yr with interest compounded monthly.
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406/26. For compound interest accounts, the amount A accumulated or due depends on the principal p, interest rate r, number of compoundings per year n, and the time t in years according to the formula A = p(1+r / n)nt.
What principal should be deposited at 8.375% compounded monthly to ensure the account will be worth $ 20,000 in 10 yr?
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406/28. For compound interest accounts, the amount A accumulated or due depends on the principal p, interest rate r, number of compoundings per year n, and the time t in years according to the formula A = p(1+r / n)nt.
Compound interest: As a follow­ up experiment ( see Exercise 27), David invests $ 10 in an account paying 12% interest compounded 10 times per year for 10 yr, and another $ 10 in an account paying 10% interest compounded 12 times per year for 10 yr. Which produces the better investment— more compounding periods or a higher interest rate?
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