To find the amount in the account after four years with continuous compounding, we can use the formula for continuously compounded interest. The formula is given by:
[tex]\[ A = P \cdot e^{(rt)} \][/tex]
where:
- [tex]\( A \)[/tex] is the amount of money accumulated after n years, including interest.
- [tex]\( P \)[/tex] is the principal amount (the initial amount of money).
- [tex]\( r \)[/tex] is the annual interest rate (in decimal form).
- [tex]\( t \)[/tex] is the time the money is invested for, in years.
- [tex]\( e \)[/tex] is the base of the natural logarithm, approximately equal to 2.71828.
Given the values:
- [tex]\( P = 2200 \)[/tex] (initial amount)
- [tex]\( r = 0.05 \)[/tex] (annual interest rate)
- [tex]\( t = 4 \)[/tex] (time in years)
we can substitute these values into the formula.
Substituting the values, we get:
[tex]\[ A = 2200 \cdot e^{(0.05 \cdot 4)} \][/tex]
First, we calculate the exponent:
[tex]\[ 0.05 \cdot 4 = 0.20 \][/tex]
Now our equation becomes:
[tex]\[ A = 2200 \cdot e^{0.20} \][/tex]
Then we compute [tex]\( e^{0.20} \)[/tex]:
[tex]\[ e^{0.20} \approx 1.22140 \][/tex]
So the equation now is:
[tex]\[ A = 2200 \cdot 1.22140 \][/tex]
Multiplying these values together, we get:
[tex]\[ A = 2687.0860679523735 \][/tex]
To find the final amount in the account after four years, we should round this amount to the nearest cent (two decimal places):
[tex]\[ A \approx 2687.09 \][/tex]
Therefore, the amount in the account after four years is approximately \$2687.09.
