455 4.2 Exponential Functions In the formula for compound interest A = P a1 + r nb t n , A is sometimes called the future value and P the present value. A is also called the compound amount and is the balance after interest has been earned. EXAMPLE 8 Finding Present Value Becky must pay a lump sum of $6000 in 5 yr. (a) What amount deposited today (present value) at 3.1% compounded annually will grow to $6000 in 5 yr? (b) If only $5000 is available to deposit now, what annual interest rate is necessary for the money to increase to $6000 in 5 yr? SOLUTION (a) A = P a1 + r nb t n Compound interest formula 6000 = P a1 + 0.031 1 b 5112 Let A = 6000, r = 0.031, n = 1, and t = 5. 6000 = P11.03125 Simplify. P = 6000 11.03125 Divide by 11.03125 to solve for P. P ≈5150.60 Use a calculator. If Becky leaves $5150.60 for 5 yr in an account paying 3.1% compounded annually, she will have $6000 when she needs it. Thus, $5150.60 is the present value of $6000 if interest of 3.1% is compounded annually for 5 yr. (b) A = P a1 + r nb t n Compound interest formula 6000 = 500011 + r25 Let A = 6000, P = 5000, n = 1, and t = 5. 6 5 = 11 + r25 Divide by 5000. a6 5b 1/5 = 1 + r Take the fifth root on each side. a6 5b 1/5 - 1 = r Subtract 1. r ≈0.0371 Use a calculator. An interest rate of 3.71% will produce enough interest to increase the $5000 to $6000 by the end of 5 yr. S Now Try Exercises 99 and 103. CAUTION When performing the computations in problems like those in Examples 7 and 8, do not round off during intermediate steps. Keep all calculator digits and round at the end of the process.
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