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The general exponential functiony=a(b)x, wherea is the initial amount or numbery is the final amountb is the basedoubling, b=2half-life, b=12increasing by x% (compound interest, appreciation), b=1+x100decreasing by x% (depreciation), b=1-x100x is the number of growth periodsdoubling, x=tDhalf-life, x=thcompound interest x=# of yearscompounding periodsyear Specific Formulas:Compound Interest-19050102235A=P1+in020000A=P1+inA is the amount. It includes the principal and the interestP is the original principal investedi is the interest rate per compounding periodn is the number of compounding periodsDepreciation-1905088900A=A01-in020000A=A01-inA is the amount after time. A0 is the original amounti is the % depreciation per yearn is the number of compounding periodsExponential Growth (involving a doubling period)-14287597790A=A02tD020000A=A02tDA is the total amount or numberA0 is the initial amount or number2 is the growth factort is the timeD is the doubling periodExponential Decay (involving half-life)-14287596520A=A012th020000A=A012thA is the remaining mass of the decayed materialA0 is the original mass of the material12is the decay factort is the timeh is the half-lifeNote: This is a summary, but try to understand the process of coming up with equations rather than memorize them.5505450-10922000Example 1One bacterium divides into two bacteria every 5 days.a) Initially, there are 15 bacteria. How many bacteria will there be in 10 days?b) What is the approximate growth rate per day?Example 2Zach’s parents invested $4000 in an account when he was born. The account pays interest at 6%/a, compounded quarterly. How much money will be in the account on Zach’s 18th birthday?53435253556000534352515430500Example 3Archaeologists use carbon-14 dating to estimate the age of relics. All living organisms contain non-radioactive carbon, carbon-12, and radioactive carbon, carbon-14. When an organism dies, the amount of carbon-12 remains the same, but carbon-14 decays exponentially. The half-life of carbon-14 is about 5370 years. Bart finds some wood and pottery in a cave. The pottery is thought to be about 8055 years old. Bart checks the age of the pottery by carbon dating the wood. If the wood is the same age as the pottery, how much carbon-14 should be in the wood? Express your answer as a percent, to the nearest tenth.Example 4 Yeast cells duplicate every 30 minutes. If an initial culture contains 75 cells how many will there be 4 hours later?Example 5 A colony of 50 bacteria results in 3200 after 2 hours. What was the doubling period? Example 6 If you invested $ 5000 for 6 years at 9% compounded semi-annually, how much would you have? Example 7 How long would it take $4000 to grow to $10 000 if it was invested at 8% compounded quarterly? Example 8 If radon has a half-life of 4 days, how much would be left of a 100g sample after 16 days?Example 9 Iodine-131 is a rare isotope. If 320 mg of iodine-131 is stored for 40 days and at the end of which time only 10 mg remain, what is its half-life? Thought-Provoking QuestionsOn a stormy day in 1988, a woman, named Sophie Smyth, walked into a Museum London with a piece of garment and claimed that she had found the linen once worn by Jesus Christ about 2000 years ago. That night, lab results revealed that the amount of carbon-14 within the garment was about 92% of the radioactivity associated with living plant materials. Given that the half-life of carbon-14 is about 5730 years, was Sophie telling the truth? A homicide detective is called to the scene of a crime where a dead body has just been found. He arrives on the scene at 9:30 pm. Immediately, the temperature of the body is taken and is found to be 80o F. The detective checks the programmable thermostat and finds that the room has been kept at a constant 68 o F. After evidence from the crime scene is collected, the temperature of the body is taken at 11:30 pm once more and found to be 78 o F. If the normal human body temperature is about 98 o F, approximately when was the victim murdered? (Hint: Newton’s Law of Cooling)Practice QuestionsInflation is causing things to cost roughly 2% more per year.A bag of milk costs approximately $3.75 now. Estimate its cost in 5 years.A movie ticket costs $11.99 now. If inflation continues at 2% per year, when will the ticket cost $15.00?How long ago did the movie ticket cost $4.25?A bacteria colony grows at a rate of 15%/h.In how many hours will the colony triple in size?In 10h, the bacteria population grows to 1.3x10?. How many bacteria were there initially?Misha drops a small rubber bouncy ball from a height of 6m onto a hard surface. After each bounce, the ball rebounds to 60% of the previous maximum height.Determine the height of each of the first 5 bounces.Graph and carefully label the relation.Create an equation to model the height of the ball as it bounces. What domain are you using?Estimate the height after 12 bounces from your graph. Verify this using your equation.Use the graph to estimate when the ball’s maximum height will be 20 cm. Verify using your equation.A population, P, is increasing exponentially. At time t=0, the population is 35 000. In 10 years, the population is 44400.Find in Using the value of that you calculated, write an equation that models the population, P, after t years.Using your equation, find when the population reaches 100 000. Which is the best investment if the money in each case is invested for three years?$5000 at 8%/a, compounded monthly$5000 at 8.2%/a, compounded annually $5000 at 8.1%/a, compounded semiannuallyAfter an accident at a nuclear power plant, which caused a radiation leak, the radiation level at the accident site was 950 R (roentgens). Five hours later, the radiation level was 800 R. Radiation levels decay exponentially. Calculate the decay rate.Matt bought a new car for $35 000 and sold it five years later to Martin for $20 000. Assume that the value of the vehicle depreciates exponentially at a rate of 12%/year. Who got ripped off? By how much?Describe three different techniques for solving 5x=90Determine the point of intersection between the graphs of y=542xand y=426x. Round your answer to two decimal places.When you drink coffee, tea, or hot chocolate, or eat a chocolate bar, your body absorbs chemicals from these foods. One chemical is caffeine. The amount of caffeine in your bloodstream follows an exponential pattern over time. After eating a food with caffeine, the highest level of caffeine in the bloodstream occurs in 15 min to 45 min. Then the level of caffeine begins to fall. In adult humans, the half-life for caffeine varies according to many factors.Adult nonsmoker: 5 h to 6 hAdult smoker: 3.5 hWoman who is six months pregnant: 10 h to 18 hNewborn baby: 100 h8-month old baby: 4 h6 year old to 10 year old child: 2 h to 3 hThe amount of caffeine in different foods varies as well:Coffee: 100 mg to 150 mgTea: 50 mg to 75 mgHot chocolate: 50 mgCola: 35 mg to 55 mgChris is an adult, smoker and drinks a cup of coffee. How much caffeine would be in Chris’s bloodstream seven hours after drinking the coffee?Answers:1) a. $4.14b. i) approximately 11 years, 4 months ii) 52 years; 2) a. approx. 8 hours, b. 3213) a. 6m, 3.6m, 2.16m, 1.30m, 0.78m, 0.47m, c. h=60.6n (h=height, n=#of bounces) the domain is n∈N (natural numbers not real numbers!) d. 0.013m, e. Between 6th and 7th bounce 4) a. 1.02, b. P=35 0001.02tc. t=44.14 5) “a” 6) 0.966 7) Matt overcharged Martin $15308) log both sides, guess and check, turn to logarithmic form9) 0.1610) 31.25mgtormyrcasthis hen ith living tim murdered? has found the linen worn by Jesus Christ 1988 years ago. ................
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