If $15 000 is invested at 10% per year, in approximately how many years will the investment double?Question 11Select one:7.3 years10.6 years14.8 years8.4 years

1. Break Down the Problem

To find out how many years it will take for an investment to double, we can use the formula for compound interest or the rule of 72. In this case, we will use the formula for compound interest:

$A = P(1 + r)^t$

where:

• $A$ is the amount of money accumulated after n years, including interest.
• $P$ is the principal amount (the initial amount of money).
• $r$ is the annual interest rate (decimal).
• $t$ is the number of years the money is invested for.

We need to find $t$ when $A = 2P$.

2. Relevant Concepts

Since we want to double the investment, we can set up the equation:

$2P = P(1 + r)^t$

We can simplify this to:

$2 = (1 + r)^t$

Given $r = 0.10$ (10%), we substitute this into the equation:

$2 = (1 + 0.10)^t$

3. Analysis and Detail

Now let's solve the equation:

$2 = (1.10)^t$

To solve for $t$, we need to take the logarithm of both sides:

$\log(2) = t \log(1.10)$

Rearranging gives:

$t = \frac{\log(2)}{\log(1.10)}$

Now we can calculate the values:

$\log(2) \approx 0.3010$ $\log(1.10) \approx 0.0414$

Substituting these values back into the equation:

$t \approx \frac{0.3010}{0.0414} \approx 7.27$

4. Verify and Summarize

The calculated value of $t$ (approximately 7.27) suggests that it will take just over 7 years for the investment to double, which is closest to 7.3 years.

Final Answer

The investment will double in approximately 7.3 years.