Question

At the end of t years, the future value of an investment of $9,000 in an account that pays 9% APR compounded monthly is S = 9,000(1 + 0.09/12)^(12t) dollars. Assuming no withdrawal or additional deposits, how long will it take for the investment to reach $27,000?

What is the value of t when the investment reaches $27,000?
a) 2 years
b) 3 years
c) 4 years
d) 5 years

Answer 1

As it takes approximately c) 4 years for the $9,000 investment to reach $27,000.

Step-by-step explanation:

The given formula for the future value of an investment is
`\(S = 9,000 * \left(1 + (0.09)/(12)\right)^(12t)\)`. We are asked to find the time
`(\(t\))` it takes for the investment to reach $27,000. Setting
`\(S\)` to $27,000, we have the equation
`\(27,000 = 9,000 * \left(1 + (0.09)/(12)\right)^(12t)\)`. To solve for
`\(t\)`, we can simplify the equation:

`\[3 = \left(1 + (0.09)/(12)\right)^(12t)\]`

Taking the natural logarithm (ln) of both sides to isolate
`\(t\)`, we get:

`\[ln(3) = 12t * ln\left(1 + (0.09)/(12)\right)\]`

Now, solving for
`\(t\)`:

`\[t = (ln(3))/(12 * ln\left(1 + (0.09)/(12)\right))\]`

Calculating this expression gives us
`\(t \approx 4\),` confirming that it takes approximately 4 years for the investment to reach $27,000. Therefore, the correct answer is (c) 4 years.

In summary, by applying the compound interest formula and logarithmic functions, we find that after around 4 years, the investment grows to $27,000. This result aligns with the option (c) in the given choices, making it the correct answer.