# Applications of Exponential Functions

## Key Questions

• Example:

John tells you a secret. You see no harm in telling Bob and Rob .

After this, 4 people know the secret (John, you Bob and Rob). Suppose that both Bob and Rob decide to tell the secret to two new people.

After the third "round" of indiscretion, eight people will know the secret.

If this pattern of spreading the secret continues, how many people will know the secret after 10 such "rounds"?

• If half-life of a certain quantity $Q \left(t\right)$ is $h$, then we can write

$Q \left(t\right) = Q \left(0\right) {\left(\frac{1}{2}\right)}^{\frac{t}{h}}$.

Example

Suppose that a certain radioactive substance has a half-life of 20 years. If you initially have 100 g of this substance, then find the quantity function $Q \left(t\right)$ of this substance after $t$ years.

Since $Q \left(0\right) = 100$ and $h = 20$, we have

$Q \left(t\right) = 100 {\left(\frac{1}{2}\right)}^{\frac{t}{20}}$.

I hope that this was helpful.