•HALF-LIFE•COMPOUND INTEREST

Exponential Expressions

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:

€

y = abx → y = a1

2

⎛

⎝ ⎜

⎞

⎠ ⎟x

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?

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?

The Equation

€

y = a1

2

⎛

⎝ ⎜

⎞

⎠ ⎟

1h t( )

OR y = a1

2

⎛

⎝ ⎜

⎞

⎠ ⎟

1λ t( )

Try This

Start with 100 grams of Iodine-131Half-life of about 8 days30 hours of time pass

How much Iodine-131 remains?

the math…

€

y = a1

2

⎛

⎝ ⎜

⎞

⎠ ⎟

1h( ) t

→ y =1001

2

⎛

⎝ ⎜

⎞

⎠ ⎟

18days( )30hours

y =1001

2

⎛

⎝ ⎜

⎞

⎠ ⎟

1192hours( )30hours

→ y =1001

2

⎛

⎝ ⎜

⎞

⎠ ⎟

30192( )

y =1001

2

⎛

⎝ ⎜

⎞

⎠ ⎟

30192( )

→ y ≈100(.8973545375)

y ≈ 89.73545375 About 89.73 g of I-131 remains

Simple Interest

Simple Interest:A=P(1+r) A = Amount after interest

appliedP = Principle: The initial

amount ofmoney lent or borrowed

r = interest rate (usually given in

percent, so you’ll have toconvert to decimal

form).

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)?

€

A = P(1+ r) → A = $100(1+ 0.05)

A = $100(1.05) → $105

Compound Interest

€

A = P 1+r

n

⎛

⎝ ⎜

⎞

⎠ ⎟nt

A = Amount after investmentP = Principle: Initial amount

invested/borrowedr = 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!

This is a form of exponential growth

Parent Function:

€

A = P 1+r

n

⎛

⎝ ⎜

⎞

⎠ ⎟nt

€

y = abx

•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.

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.

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.

€

A = P 1+r

n

⎛

⎝ ⎜

⎞

⎠ ⎟nt

→ A = $5000 1+.025

4

⎛

⎝ ⎜

⎞

⎠ ⎟4 5( )

A = $50004

4+

.025

4

⎛

⎝ ⎜

⎞

⎠ ⎟20

→ A = $50004.025

4

⎛

⎝ ⎜

⎞

⎠ ⎟20

A = $5000 4.025 /4( )20

→ A ≈ $5663.54

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!

Continuous Compounding

€

A = Pert 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.

€

1+1

x

⎛

⎝ ⎜

⎞

⎠ ⎟x

, x →∞

€

π

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%:

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%:

€

A = Pert → A = $5000e 0.025( )5

A = $5000e.125 → A ≈ $5000(1.133148)

A ≈ $5665.74