Certainly! Let's work through this step-by-step to derive the equation for the exponential growth of the money you deposited in the bank.
1. Initial Amount (Principal):
You start with an initial deposit, also known as the principal. In your case, this amount is [tex]$1,000.
2. Growth Rate:
The money grows at a rate of 6% per year. To express this as a decimal, we use 0.06.
3. Exponential Growth Formula:
The general formula for exponential growth is:
\[
A(t) = P \times (1 + r)^t
\]
where:
- \(A(t)\) is the amount of money at time \(t\),
- \(P\) is the principal amount (initial deposit),
- \(r\) is the growth rate per period,
- \(t\) is the number of periods (years, in this case).
4. Substitute the Values:
Given \(P = 1000\) and \(r = 0.06\), we substitute these values into the formula:
\[
A(t) = 1000 \times (1 + 0.06)^t
\]
5. Simplify the Equation:
Combine the terms inside the parentheses:
\[
A(t) = 1000 \times 1.06^t
\]
Thus, the equation representing the exponential growth of your $[/tex]1,000 deposit at a 6% annual growth rate is:
[tex]\[
A(t) = 1000 \times 1.06^t
\][/tex]
This equation tells you how much money you will have in the account after [tex]\(t\)[/tex] years.
