Answer:
4% (nearest whole number)
Step-by-step explanation:
Continuous Compounding Formula
[tex]\large \text{$ \sf A=Pe^{rt} $}[/tex]
where:
- A = Final amount.
- P = Principal amount.
- e = Euler's number (constant).
- r = Annual interest rate (in decimal form).
- t = Time (in years).
Given values:
- A = $1,822
- P = $1,000
- t = 15 years
Substitute the given values into the formula and solve for r:
[tex]\implies \sf 1822=1000 \cdot e^{15r}[/tex]
[tex]\implies \sf \dfrac{1822}{1000}=e^{15r}[/tex]
[tex]\implies \sf 1.822=e^{15r}[/tex]
[tex]\implies \sf \ln 1.822=\ln e^{15r}[/tex]
[tex]\implies \sf \ln 1.822=15r \ln e[/tex]
[tex]\implies \sf \ln 1.822=15r[/tex]
[tex]\implies \sf r=\dfrac{\ln 1.822}{15}[/tex]
[tex]\implies \sf r=0.039995653...[/tex]
To convert into a percentage, multiply by 100:
[tex]\implies \sf r=0.039995653... \times 100[/tex]
[tex]\implies \sf r=3.99995653...\%[/tex]
[tex]\implies \sf r=4\%\; (nearest\;whole\;number)[/tex]
Therefore, the interest rate is 4% (nearest whole number).
