3.5 Exponential Growth & Decay Applications that Apply to Me!

What real-life applications are there?

Think-Ink-Pair: Money Doubling? You have a $100.00 Your money doubles each year. How much do you have in 5 years? Show work.

Money Doubling Year 1: $100 · 2 = $200 Year 2: $200 · 2 = $400 Year 3: $400 · 2 = $800 Year 4: $800 · 2 = $1600 Year 5: $1600 · 2 = $3200

Earning Interest on You have $100.00. Each year you earn 10% interest. How much $ do you have in 5 years? Show Work.

Earning 10% results Year 1: $100 + 100·(.10) = $110

a = Principle or starting amount Growth Models: The Equation is: y = a (1+ r)t a = Principle or starting amount r = percent increase t= time

Using the Equation $100.00 10% interest 5 years 100(1+ (.10))5 = $161.05 What could we figure out now?

Comparing Investments Choice 1 – Bank of America $10,000 5.5% interest 9 years Choice 2 – Wells Fargo $8,000 6.5% interest 10 years

Choice 1 $10,000, 5.5% interest for 9 years. Equation: $10,000 (1 + .055)9   Balance after 9 years: $16,190.94

Choice 2 $8,000 in an account that pays 6.5% interest for 10 years.   Equation: $8,000 (1 + .065)10 Balance after 10 years: $15,071.10

The first one yields more money. Which Investment? The first one yields more money. Choice 1: $16,190.94 Choice 2: $15,071.10

Instead of increasing, it is decreasing. Exponential Decay Instead of increasing, it is decreasing. Formula: y = a (1 – r)t a = initial amount r = percent decrease t = Number of years

Real-life Examples What is car depreciation? Car Value = $20,000 Depreciates 10% a year Figure out the following values: After 2 years After 5 years After 8 years After 10 years

Exponential Decay: Car Depreciation Assume the car was purchased for $20,000 Depreciation Rate Value after 2 years Value after 5 years Value after 8 years Value after 10 years Formula: y = a (1 – r)t a = initial amount r = percent decrease t = Number of years

What Else? What happens when the depreciation rate changes. What happens to the values after 20 or 30 years out – does it make sense? What are the pros and cons of buying new or used cars.

Compound Interest A = Ending Amount P = Starting (principle) Amount r = interest rate t= number of years n = Number of times the interest is compounded in a year. *annually = 1 *quarterly = 4 *semi-annually = 2 *monthly = 12

Compound Interest Example: You deposit $5000 into a savings account that earns 3% annual interest. If no other money is deposited, what is your balance after 4 years if it is compounded… Semi-annually: Quarterly: Monthly: Daily: Annually: