Question

Suppose that Mike invests $5000 into an account that pays 12% interest compounded

continuously. He wants to know how many years it will take for his investment to double and be worth
$10,000. Can Mike solve the following equation to find his answer? Why or why not?
2 = e 120. Solve the equation listed in the question above for t. First find the exact answer. Then
approximate your answer to three decimal places

Answer 1

t = log(2)/0.12 years (exact value)

= 5.776 years (to three decimal places)

Explanation:

The future value of an amount P at continuous interest rate i% for t years can be calculated using the continuous interest formula:

FV = Pe^((i)t)

where e is the Euler's constant = 2.7182818284....

We're given

10000 = 5000 e^((0.12)t)

simplify

2 = e^(0.12t) ........................ see below

[ please compare if this is the same as the one shown in the original question, the formula posted as "2 = e 120" does not look right, probably because of typographical difficulties, then answer if the equation given in the question can be used to solve the problem]

take (natural) log using the power property of logarithms

log(2) = 0.12t log(e)

using natural log, log(e) = 1

log(2) = 0.12t

simplify

t = log(2)/0.12 years (exact value)

To obtain the time (in years)

t = log(2)/0.12 = 0.6931/0.12 = 5.7762 years

Check using the rule of 72 (approximate)

72/12(%) = 6,

Since continuous interest accumulates interest faster, so 5.8 years sound reasonable.