1. HALF-LIFE COMPOUND INTEREST Exponential Expressions

2. Half-Life Exponential Decay.  Meaning that your “b” value is less than one (and of course greater than zero).  “b” is a fraction Parent Function:

3. What does half-life look like? If you had 50 grams of a compound and half of it changes, how much of the original compound remains? If you look at the remaining material and half of it changes, how much of the original compound remains?

4. Try This: A particular elemental compound has a 20 day half- life.  If 40 days pass:  How many half-lives have passed?  How much of the sample would remain?  If 60 days pass:  How many half-lives have passed?  How much of the sample would remain?  If 70 days pass:  How many half-lives have passed?  How much of the sample would remain?

5. The Equation

6. Try This Start with 100 grams of Iodine-131 Half-life of about 8 days 30 hours of time pass How much Iodine-131 remains?

7. the math… About 89.73 g of I-131 remains

8. Simple Interest Simple Interest: A=P(1+r)A = Amount after interest applied P = Principle: The initial amount of money lent or borrowed r = interest rate (usually given in percent, so you’ll have to convert to decimal form).

9. Example of Simple Interest You borrow $100.00. You have to pay 5% interest to borrow the money. How much money will you have to pay to pay back the loan (with interest)?

10. Compound Interest A = Amount after investment P = Principle: Initial amount invested/borrowed r = interest rate (again, usually given in percent form – must convert to decimal form n = number of times the interest is applied to the principle and then the amount is added back to the original = compounding t = the length of the investment/loan This is how credit card companies make their money!!! The interest is added back to the principle and this is treated as the NEW amount that has been borrowed!

11. This is a form of exponential growth Parent Function: The Principle is just like the “a” value in the parent function The expression (1+r) is another way to express the growth factor “b” The only difference is the addition of “n” – the number of times the expression is compounded.

12. Compound Interest Example: A relative has decided to give you $5000, but says you must invest the money for at least five years. After some research, you can find an interest rate of 2.5%. Compounded quarterly.

13. Compound Interest Example: A relative has decided to give you $5000, but says you must invest the money for at least five years. After some research, you can find an interest rate of 2.5%. Compounded quarterly.

14. What if the compounding is continuous Instead of compounding annually (once a year), semi-annually (twice a year), monthly (12 times a year), daily (365 times a year) and so on… What if the compounding happened every possible moment throughout the investment / growth. This is what happens in natural systems by the way. Growth isn’t incremental, it’s continuous!

15. Continuous Compounding A is still the amount accrued after investment P is still the principle amount r is still the interest rate in decimal form t is still the time of investment The natural number e is a value that is found when a system is allowed to compound infinitely: e can be approximated to 2.718281828 Treat e like. It is an irrational number that has a special use. In this case, it is used in exponential expressions where growth is continuous.

16. Example on Continuous Compounding From the previous example of $5000 invested for 5 years. This time however, it will be continuously compounded at 2.5%:

17. Example on Continuous Compounding From the previous example of $5000 invested for 5 years. This time however, it will be continuously compounded at 2.5%: