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Question

If P dollars is invested in a savings account that pays 5 percent annual interest, compounded continuously, in how many years will the account value be equal to 2P dollars?

Answer 1

To solve this problem, we can use the formula for compound interest:

A = P * e^(rt)

Where:
A is the final account value
P is the initial principal (dollars)
e is the mathematical constant approximately equal to 2.71828
r is the annual interest rate (decimal form)
t is the time in years

In this case, we want to find the time it takes for the account value to be equal to 2P dollars. So we have:

2P = P * e^(0.05t)

Divide both sides of the equation by P:

2 = e^(0.05t)

Take the natural logarithm (ln) of both sides:

ln(2) = 0.05t * ln(e)

Since ln(e) equals 1, the equation simplifies to:

ln(2) = 0.05t

Now we can divide both sides of the equation by 0.05:

ln(2) / 0.05 = t

Using a calculator, we find:

t ≈ 13.86

Therefore, it will take approximately 13.86 years for the account value to be equal to 2P dollars.