QuestionBrian deposited $\$ 2000$ into an account with a $5.6 \%$ annual interest rate, compounded semiannually. Assuming that no withdrawals are made, how long will it take for the investment to grow to $\$ 3026$ ?
Do not round any intermediate computations, and round your answer to the nearest hundredth.

Studdy Solution

STEP 1

1. The principal amount deposited by Brian is $2000.<br />2. The annual interest rate is 5.6%.<br />3. The interest is compounded semiannually.<br />4. The future value of the investment is$ 3026.
5. We are trying to find the time it takes for the investment to grow to $3026.

STEP 2

1. Identify the formula for compound interest.
2. Define the variables.
3. Substitute the known values into the formula.
4. Solve for the unknown variable, which is time.

STEP 3

Identify the formula for compound interest.
The compound interest formula is:
$A = P \left(1 + \frac{r}{n}\right)^{nt}$
where: - $A$ is the future value of the investment/loan, including interest. - $P$ is the principal investment amount (the initial deposit or loan amount). - $r$ is the annual interest rate (decimal). - $n$ is the number of times that interest is compounded per year. - $t$ is the number of years the money is invested or borrowed for.

STEP 4

Define the variables.
Given: - $A = 3026$ - $P = 2000$ - $r = 5.6\% = 0.056$ - $n = 2$ (since the interest is compounded semiannually)
We need to find $t$ .

STEP 5

Substitute the known values into the formula.
$3026 = 2000 \left(1 + \frac{0.056}{2}\right)^{2t}$
Simplify the expression inside the parentheses:
$3026 = 2000 \left(1 + 0.028\right)^{2t}$ $3026 = 2000 (1.028)^{2t}$

STEP 6

Solve for the unknown variable, which is time.
First, divide both sides by 2000 to isolate the exponential expression:
$\frac{3026}{2000} = (1.028)^{2t}$ $1.513 = (1.028)^{2t}$
Take the natural logarithm of both sides to solve for $t$ :
$\ln(1.513) = \ln((1.028)^{2t})$
Using the logarithmic identity $\ln(a^b) = b \ln(a)$ , we have:
$\ln(1.513) = 2t \cdot \ln(1.028)$
Solve for $t$ :
$t = \frac{\ln(1.513)}{2 \cdot \ln(1.028)}$
Calculate the value:
$t \approx \frac{0.414}{2 \cdot 0.0277}$ $t \approx \frac{0.414}{0.0554}$ $t \approx 7.47$
Therefore, it will take approximately $\boxed{7.47}$ years for the investment to grow to $3026.

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