Question

$25,000 is deposited into an account that pays 6.75% interest compounded

continuously. How long will it take for the money to double in value?


Show steps please!!!!

Answer 1

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

A = Pe^(rt)

where A is the amount of money in the account after t years, P is the initial amount invested, r is the annual interest rate as a decimal, and e is the mathematical constant e.

We want to find out how long it will take for the money to double, which means we want to find the value of t that satisfies:

2P = Pe^(rt)

Dividing both sides by P, we get:

2 = e^(rt)

Taking the natural logarithm of both sides, we get:

ln(2) = rt ln(e)

But ln(e) = 1, so we can simplify to:

ln(2) = rt

Solving for t, we get:

t = ln(2) / r

Now we can plug in the values from the problem:

P = $25,000

r = 0.0675 (since 6.75% is 0.0675 as a decimal)

t = ln(2) / 0.0675

t ≈ 10.28 years

Therefore, it will take approximately 10.28 years for the money to double in value.