Question

How many years will it take for an initial investment of $ 10,000 to grow to $ 25,000 ?

Assume a rate of interest of 15 % compounded continuously. It will take about years for the

Answer 1

It will take approximately 6.11 years for an initial investment of $10,000 to grow to $25,000 at a 15% interest rate compounded continuously. The calculation is based on the formula
`\( t = (\ln(A/P))/(r) \)`.

Step-by-step explanation:

The formula for compound interest compounded continuously is given by the formula:

`\[A = P \cdot e^(rt),\]`

where:

(A) is the future value of the investment,

(P) is the principal amount (initial investment),

(e) is the mathematical constant approximately equal to 2.71828,

(r) is the annual interest rate (as a decimal),

(t) is the time the money is invested for in years.

In your case:

(P = $10,000) (initial investment),

(A = $25,000) (desired future value),

(r = 0.15) (15% interest rate as a decimal),

(e\) is the mathematical constant,

(t\) is the time we want to find.

The formula to find \(t\) is:

`\[t = (\ln(A/P))/(r),\]`

where
`\(\ln\)` is the natural logarithm.

Now, let's plug in the values:

`\[t = (\ln\left((25,000)/(10,000)\right))/(0.15).\]`

Calculate this expression to find the time (t). Using a calculator or software, you get:

`\[t \approx (\ln(2.5))/(0.15) \approx (0.91629)/(0.15) \approx 6.10793`\]

So, it will take approximately \(6.11\) years for the initial investment of $10,000 to grow to $25,000 at an interest rate of 15% compounded continuously.