Question

You invest $5,000 into an account where interest compounds continuously at 3.5%. How long will it take your money to double? Round answer to nearest year.

Answer 1

Answer:

20 years

Step-by-step explanation:

Continuous Compounding Formula

$\large \sf A=Pe^{rt}$

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:

$\sf \implies 10000=5000e^{0.035t}$

$\sf \implies \dfrac{10000}{5000}=e^{0.035t}$

$\sf \implies 2=e^{0.035t}$

$\sf \implies \ln 2=\ln e^{0.035t}$

$\sf \implies \ln 2=0.035t\ln e$

$\sf \implies \ln 2=0.035t(1)$

$\sf \implies \ln 2=0.035t$

$\sf \implies t=\dfrac{\ln 2}{0.035}$

$\implies \sf t=19.80420516...$

Therefore, it will take 20 years (to the nearest year) for the initial investment to double.