Compound interest formula P = the principal (the initial amount) A= r=annual interest rate (expressed as a decimal) n=number of interest periods per year (see the table below for more information) t=number of years P is invested A=amount after t years If investment interest rate is compounded monthly, then n = 12 If investment interest rate is compounded quarterly, then n = 4 If investment interest rate is compounded semi-annually, then n = 2 If investment interest rate is compounded annually, then n = 1 Continuously compounded interest formula A= P = the principal (the initial amount) r=annual interest rate (expressed as a decimal) t=number of years P is invested A=amount after t years Law of growth(or decay) formula q=q(t)=q 0 e rt (i.e q changes instantaneously at a rate proportional to its current value) Where : q 0 =the value of q at time t=0 (that is, q0 is the initial amount) If r>0 then r is called the growth rate of q(the growth constant) If r<0 then r is called the decay rate of q (the decay constant)
Practice exercises 1. $1000 is invested at a rate of 11% per year compounded monthly. (a) Find the principal after 1 month (b) Find the principal after 1 year. 2. If a savings fund pays interest at a rate of 10% per year compounded semiannually, how much money invested now will amount to $4500 after 1 year? (Round the answer to the nearest cent.) 3. If P dollars is deposited in a savings account that pays interest at a rate of r % per year compounded continuously, find the balance after t years. (Round your answer to the nearest cent.) 4. The 1980 population of the United States was approximately 231 million, and the population has been growing continuously at a rate of 1.03% per year. Predict the population in the year 2023 if this growth trend continues. (Give the answer correct to two decimal places.)
5. The energy E(x) of an electron after passing through material of thickness x is given by the equation E(x) = E 0 e (-x/x 0 ), where E 0 is the initial energy and x 0 is the radiation length. (a) Express, in terms of E 0, the energy of an electron after it passes through material of thickness x 0. E(x 0 ) = (b) Express, in terms of x 0, the thickness at which the electron loses 99% of its initial energy. (Round the answer to the nearest tenth.) 6. If a certain bacteria population triples in 2 hours, determine the time T (in hours) that it takes the population to double. Enter either an exact expression or a number correct to the nearest tenth of an hour. T= 7. Money is invested with continuously compounding, and triples in time t. Express the interest rate, r, as a function of t. Interest rate as a function of tripling time= Half-life is the period of time it takes for a substance undergoing decay to decrease by half. 8. 86% of a radioactive material remains after 30 days. Part 1: Find the decay constant. IMPORTANT: Decimal answers will be marked incorrect--a decimal point can't be ANYWHERE in your answer. Enter an EXACT, symbolic answer (such as 3/2, not 1.5, or ln(2), not 0.69) Part 2: Find the time T (in days) after the initial measurement when 36% of the original amount of radioactive material remains. You may enter a symbolic answer or round your answer to the nearest whole number.
9. A certain radioactive substance with a half-life of 3200 years is used in estimating the age of relics. Part 1: Find the decay constant. Your answer must be EXACT. Answers which included decimal points ANYWHERE will be marked incorrect. Enter an EXACT, symbolic answer (such as 3/2, not 1.5, or ln(2), not 0.69 10. Two populations of bacteria are growing exponentially in separate petri dishes. The population in the first dish has growth constant 0.16 and initial population 1000. The population in the second dish has growth constant 0.07 and initial population 6000. Part 1. Find the time at which the two populations have equal size. Your answer must be EXACT--decimal points ANYWHERE will be marked incorrect (so you must convert those growth constants). Time at which the populations are equal= Part 2. What is the common value of the two populations at that time? Either enter an exact expression or round to the nearest whole number. 11. This question has two parts. Alice invests $8000 at Bob's bank and $9000 at Charlie's bank. Bob compounds interest continuously at a nominal rate of 9%. Charlie compounds interest continuously at a nominal rate of 3%.
Part 1: In how many years will the two investments be worth the same amount? Your answer must be EXACT. Answers which included decimal points ANYWHERE will be marked incorrect (so convert those interest rates to fractions!). Enter an EXACT, symbolic answer (such as 3/2, not 1.5, or ln(2), not 0.69 Part 2: When both investments are worth the same amount, how much will each be worth? IMPORTANT: Round your answer to the nearest cent. Do not include a dollar sign in your answer. 12. The graph shown below is the graph of an exponential growth curve, P = P 0e kt Part 1: Find the growth constant, k. Your answer must be exact, that is, symbolic (NO decimal places anywhere). Part 2: Find P 0. Your answer must be correct to one decimal place. (Use the exact form of k. to insure your answer is accurate.) 13. In 1974, Johnny Miller won 8 tournaments and accumulated $353,032 in official season earnings. In 1999, Tiger Woods accumulated $6,616,585 with a similar record.
(a) Suppose the MONTHLY inflation rate from 1974 to 1999 was 0.0067 (a decimal, not a percent). Use the compound interest formula with this monthly rate to estimate the equivalent value of Miller's winnings in the year 1999. (Round the answer to the nearest hundredth, and enter this value, not a formula.) (b) Find the annual interest rate needed for Miller's winnings to be equivalent in value to Woods's winnings, assuming monthly compounding. (Express your answer as a percent, and round the answer to the nearest hundredth.) (c) What type of function did you use in part (a)? ( linear, polynomial, exponential) d) What type of function did you use in part (b)?(linear, polynomial, exponential)