To determine how much money will be in the account when you retire at age 68, given that you initially invest [tex]$9,000 at age 18, we need to use the formula for continuous compounding, which is:
\[ P(t) = P_0 \cdot e^{rt} \]
where:
- \( P(t) \) is the final amount in the account after \( t \) years.
- \( P_0 \) is the initial investment, which is $[/tex]9,000.
- [tex]\( r \)[/tex] is the annual rate of return, which is 0.067 (or 6.7% per year).
- [tex]\( t \)[/tex] is the number of years the money is invested, which is 50 years.
- [tex]\( e \)[/tex] is the base of the natural logarithm, approximately equal to 2.71828.
Given these values:
- [tex]\( P_0 = 9000 \)[/tex] dollars
- [tex]\( r = 0.067 \)[/tex]
- [tex]\( t = 50 \)[/tex]
Plugging the values into the formula:
[tex]\[ P(50) = 9000 \cdot e^{0.067 \times 50} \][/tex]
Now, let's calculate the exponent:
[tex]\[ 0.067 \times 50 = 3.35 \][/tex]
So, we now have:
[tex]\[ P(50) = 9000 \cdot e^{3.35} \][/tex]
The value of [tex]\( e^{3.35} \)[/tex] is approximately 28.50273476.
Therefore:
[tex]\[ P(50) = 9000 \cdot 28.50273476 \][/tex]
Multiplying these together:
[tex]\[ P(50) = 256524.60279390553 \][/tex]
So, if you invest [tex]$9,000 when you are 18 and leave it alone for 50 years until you retire at 68, the amount of money that will be in the account, assuming continuous compounding, will be approximately $[/tex]256,524.60.
