Answer:
20 years
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:
- A = $10,000
- P = $5,000
- r = 3.5% = 0.035
Substitute the given values into the formula and solve for t:
[tex]\sf \implies 10000=5000e^{0.035t}[/tex]
[tex]\sf \implies \dfrac{10000}{5000}=e^{0.035t}[/tex]
[tex]\sf \implies 2=e^{0.035t}[/tex]
[tex]\sf \implies \ln 2=\ln e^{0.035t}[/tex]
[tex]\sf \implies \ln 2=0.035t\ln e[/tex]
[tex]\sf \implies \ln 2=0.035t(1)[/tex]
[tex]\sf \implies \ln 2=0.035t[/tex]
[tex]\sf \implies t=\dfrac{\ln 2}{0.035}[/tex]
[tex]\implies \sf t=19.80420516...[/tex]
Therefore, it will take 20 years (to the nearest year) for the initial investment to double.
