interest and stuff

A(t)=Ao(1+r)^t
-Used when compounding.
Ao=starting amount
r=rate
t= amount of time compounding in given time span.

A(t)=A0b^t/k
-Used when things are doubling, tripling, halving, etc.
Ao=starting amount
b=doubling, tripling, etc.
t=amount of time
k=how long it takes to double, triple, half, etc.

Rule of 72- to find how long it takes for something to double then find 72/r
r is not a decimal here

**Only used when compounding continuously
P(t)=Poe^rt
Po=starting amount
e=on ln button
r=rate
t=time

Example 1:
If $1,000 is invested in a savings account that is compounding continuously at .8% for 6 years how much money did you gain?
Compounding continuously
Pₒ = $1,000
e = e
r = .008
t = 72 months
P(t) = 1000(e^(.008*72))
= $1,778.91 – 1,000 = $778.91

Example 2:
The half-life of a radioactive isotope is 4 days. If 3.2 kg are present now, how much will be present after:
a) 4 days
A(t)=Aob^t/k
A(t)=3.2(1/2)^(4/4)
=1.6kg

b) 8 days
A(t)=Aob^t/k
=3.2(1/2)^(8/4)
=0.8 kg

c) 20 days
A(t)= Aob^t/k
A(t)=3.2(1/2)^(20/4)
=0.1 kg

d) t days
A(t)=Aob^t/k
A(t)=3.2(1/2)^t/4

Example 3:
How long does it take for any given amount to double at a rate of 8% increase per year?
This is a rule of 72 problem.
72 ÷ 8 = 9 years