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Exponential Growth/Decay Review
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Learning Targets I can write an exponential equation to model exponential growth/decay. I can write an exponential equation to model an investment or a loan. I can solve an exponential equation for the principal, rate, or amount (given the time).
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Equations you should know…
A - Starting Amount r - rate (percent as a decimal) x - time P – Principal (Initial investment/loan) r – rate (percent as a decimal) n – number of times the interest is compounded per year t – number of years.
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y=126.43 Answer: Ticket price will be $126.43 in 2014.
Example 1: Exponential Growth The price of Minnesota Vikings tickets were $68.77 in 2005 and has grown 7% per year. What will the ticket prices be in 2014? Starting Amount A – 68.77 Rate r – (7% as a decimal) Time x – 9 years ( ) y – variable we’re solving for Solve for the missing variable. Enter into calculator. y=126.43 Answer: Ticket price will be $ in 2014.
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r = 0.023 Answer: Decaying by 2.3% per hour
Example 2: Exponential Decay A petri dish started with 650,000 bacteria. Find the rate the bacteria are decreasing by per minute if there are 228,000 after 45 minutes. Starting Amount A – 650,000 Rate r – is the variable we solve for Time x – 45 minutes y – 228,000 bacteria after 45 minutes Solve for the missing variable. Divide both sides by 650,000 Raise both sides to the reciprocal power to undo the exponent. Subtract 1 on both sides. Divide by -1 on both sides. r = 0.023 Answer: Decaying by 2.3% per hour
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Compound Interest What is Compound Interest?
Addition of Interest to an account Note: If interest is compounded monthly, interest is added to your account once a month (12 times a year) Also common… Semi-Annually – 2 times a year Quarterly – 4 times a year Weekly – 52 times a year Daily – 360 times a year
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Compound Interest Formula
A: amount of money over a given time P: Principal amount (Starting amount) r: YEARLY interest rate (as decimal) N: Frequency of compound t: years
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1 48 1 48 Answer: 3% annual interest
Example 3: Investment (Solving for the interest rate) You deposit $89 in an account compounded monthly. In 4 years, you have $ Find the annual percent interest. Write an exponential equation to model this investment 100.33= 𝑟 ∙4 Principle P– 89.00 Rate r – is the variable we solve for Time t – 4 years A – $ (Amount of after 4 years) Compounded N– 12 Solve for the missing variable. 100.33= 𝑟 ∙4 Divide both sides by 89 = 𝑟 Raise both sides to the reciprocal power to undo the exponent. = 1+ 𝑟 12 Subtract 1 on both sides. = 𝑟 12 Multiply by 12 on both sides. ∙ ∙12 𝑟=0.03 Answer: 3% annual interest
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Answer: Approximately $600 invested 6 years ago.
Example 4: Investment (Solving for the principle) You have a savings account that is being compounded quarterly with a 4.5% annual interest. The account currently has $785. How much did you invest 6 years ago when the account was opened? Write an exponential equation to model this investment 785=𝑃 ∙6 Principle P– is the variable we solve for Rate r – (4.5% as a decimal) Time t – 6 years A – $785 (Amount of after 6 years) Compounded N– 4 (quarterly) Solve for the missing variable. 785=1.31𝑃 Simplify =1.31 using calculator. 785=1.31P Divide both sides by 1.31 𝑝=600.16 Answer: Approximately $600 invested 6 years ago.
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New Material All we are doing that is ‘new’ is getting you to solve for the missing time. However, we have been practicing it for the last few days!!
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Example 5: Population (Solving for the time) A town has a population of 500 people. The rate is growing by 15% per year. When will the town double the population. Write an exponential equation to model this investment 1000= 500(1+.15) 𝑥 A – Starting amount (500 people) Rate r – .15 which is 15% Time t – missing Y – ending amount (1000, which is double 500) Solve for the missing variable. 2= (1+.15) 𝑥 Divide by 500 on both sides. log 2 =𝑥 log(1.15) Log both sides and pull the variable out front. log 2 =𝑥 log(1.15) log(1.15) log(1.15) Answer: 4.95 years
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