SECTION 5.7 Financial Models 351 Determining the Time Required to Double or Triple an Investment (a) How long will it take for an investment to double in value if it earns 5% compounded continuously? (b) How long will it take to triple at this rate? Solution EXAMPLE 7 (a) If P is the initial investment and P is to double, then the amount A will be P2 . Use formula (4) for continuously compounded interest with = r 0.05. = = = = = ≈ A Pe P Pe e t t 2 2 0.05 ln2 ln2 0.05 13.86 rt t t 0.05 0.05 It will take about 14 years to double the investment. (b) To triple the investment, let = A P3 in formula (4). = = = = = ≈ A Pe P Pe e t t 3 3 0.05 ln 3 ln 3 0.05 21.97 rt t t 0.05 0.05 It will take about 22 years to triple the investment. A P r 2 , 0.05 = = Cancel the P ’s. Rewrite as a logarithm. Solve for t . A P r 3 , 0.05 = = Cancel the P ’s. Rewrite as a logarithm. Solve for t . Now Work PROBLEM 35 ‘Are You Prepared?’ Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red. 5.7 Assess Your Understanding 1. What is the interest due when $500 is borrowed for 6 months at a simple interest rate of 6% per annum? (pp. A67–A68) 2. If you borrow $5000 and, after 9 months, pay off the loan in the amount of $5500, what per annum rate of interest was charged? (pp. A67–A68) Concepts and Vocabulary 3. The total amount borrowed (whether by an individual from a bank in the form of a loan or by a bank from an individual in the form of a savings account) is called the . 4. If a principal of P dollars is borrowed for a period of t years at a per annum interest rate r, expressed as a decimal, the interest I charged is = . Interest charged according to this formula is called . 5. The is the annual simple interest rate that would yield the same amount as compounding n times per year, or continuously, after 1 year. 6. Multiple Choice The principal that must be invested now so that it will grow to a given amount in a specified time period is called the . (a) present value (b) future value (c) interest (d) effective rate 1. Now Work 1. Modeling 1.ExplainingConcepts Calculus Preview 1.InteractiveFigure Skill Building In Problems 7–14, find the amount that results from each investment. 7. $100 invested at 4% compounded quarterly after a period of 2 years 8. $50 invested at 6% compounded monthly after a period of 3 years 9. $900 invested at 3% compounded semiannually after a period of 2 1 2 years 10. $300 invested at 12% compounded monthly after a period of 1 1 2 years
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