Certainly! To determine how much \[tex]$500 invested at 8% interest compounded continuously would be worth after 3 years, we use the formula for continuous compounding:
\[ A(t) = P \cdot e^{rt} \]
Where:
- \( P \) represents the principal amount (initial investment)
- \( r \) is the annual interest rate (as a decimal)
- \( t \) is the time the money is invested for (in years)
- \( e \) is the base of the natural logarithm, approximately equal to 2.71828
- \( A(t) \) is the amount of money accumulated after time \( t \)
Given:
- Principal (\( P \)) = \$[/tex]500
- Annual interest rate ([tex]\( r \)[/tex]) = 8% or 0.08 (as a decimal)
- Time ([tex]\( t \)[/tex]) = 3 years
We substitute these values into the formula:
[tex]\[ A(3) = 500 \cdot e^{0.08 \cdot 3} \][/tex]
First, calculate the exponent:
[tex]\[ 0.08 \cdot 3 = 0.24 \][/tex]
So the formula becomes:
[tex]\[ A(3) = 500 \cdot e^{0.24} \][/tex]
Next, calculate [tex]\( e^{0.24} \)[/tex]. Using the true value obtained:
[tex]\[ e^{0.24} \approx 1.2712485753215237 \][/tex]
Thus:
[tex]\[ A(3) = 500 \cdot 1.2712485753215237 \][/tex]
Now perform the multiplication:
[tex]\[ A(3) \approx 635.6245751607024 \][/tex]
Finally, we round this value to the nearest cent:
[tex]\[ A(3) \approx 635.62 \][/tex]
Therefore, the amount that \[tex]$500 invested at an 8% interest rate compounded continuously would be worth after 3 years is approximately \$[/tex]635.62.
