Question

incorrect
your answer is incorrect.
a principal of $3800 is invested at 7% interest, compounded annually. how much will the investment be worth after 8 years?
use the calculator provided and round your answer to the nearest dollar.

Explanation:

Step1: Recall compound interest formula

The formula for compound interest is $A = P(1 + \frac{r}{n})^{nt}$, 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).
- $n$ is the number of times that interest is compounded per year.
- $t$ is the time the money is invested for in years.

Here, $P = 3800$, $r = 7\% = 0.07$, $n = 1$ (compounded annually), and $t = 8$.

Step2: Substitute values into formula

Substitute the values into the formula: $A = 3800(1 + \frac{0.07}{1})^{1\times8}$

Simplify the exponent and the fraction: $A = 3800(1 + 0.07)^{8}$

Calculate $(1 + 0.07) = 1.07$, so $A = 3800\times(1.07)^{8}$

Step3: Calculate $(1.07)^8$

First, calculate $1.07^8$. Using a calculator, $1.07^8 \approx 1.71818617$

Step4: Multiply by principal

Multiply this by the principal: $A = 3800\times1.71818617$

Calculate $3800\times1.71818617 \approx 6529.107446$

Step5: Round to nearest dollar

Round $6529.107446$ to the nearest dollar, which is $6529$. Wait, wait, let's recalculate $1.07^8$ more accurately. Let's compute step by step:

$1.07^2 = 1.1449$

$1.07^4 = (1.1449)^2 = 1.31079601$

$1.07^8 = (1.31079601)^2 = 1.718186154121$

Then $3800\times1.718186154121 = 3800\times1.718186154121$

Calculate $3800\times1.718186154121$:

$3800\times1 = 3800$

$3800\times0.7 = 2660$

$3800\times0.018186154121 \approx 3800\times0.018186154121 \approx 69.10738566$

Adding them up: $3800 + 2660 = 6460$; $6460 + 69.10738566 = 6529.10738566$. Wait, but maybe I made a mistake earlier. Wait, let's use a calculator for $3800*(1.07)^8$. Let's compute $(1.07)^8$:

Using a calculator, $1.07^8 = e^{8\ln(1.07)} \approx e^{8\times0.0676586} \approx e^{0.5412688} \approx 1.718186$

Then $3800\times1.718186 = 3800\times1.718186$

$3800\times1.7 = 6460$

$3800\times0.018186 = 3800\times0.01 + 3800\times0.008186 = 38 + 31.1068 = 69.1068$

So total is $6460 + 69.1068 = 6529.1068$, which rounds to $6529$? Wait, but maybe I miscalculated. Wait, no, let's check with another method. Wait, maybe the correct calculation is:

Wait, $1.07^8$:

Year 1: 3800*1.07 = 4066

Year 2: 4066*1.07 = 4066 + 4066*0.07 = 4066 + 284.62 = 4350.62

Year 3: 4350.62*1.07 = 4350.62 + 4350.62*0.07 = 4350.62 + 304.5434 = 4655.1634

Year 4: 4655.1634*1.07 = 4655.1634 + 4655.1634*0.07 = 4655.1634 + 325.861438 = 4981.024838

Year 5: 4981.024838*1.07 = 4981.024838 + 4981.024838*0.07 = 4981.024838 + 348.6717387 = 5329.6965767

Year 6: 5329.6965767*1.07 = 5329.6965767 + 5329.6965767*0.07 = 5329.6965767 + 373.078760369 = 5702.77533707

Year 7: 5702.77533707*1.07 = 5702.77533707 + 5702.77533707*0.07 = 5702.77533707 + 399.194273595 = 6101.96961066

Year 8: 6101.96961066*1.07 = 6101.96961066 + 6101.96961066*0.07 = 6101.96961066 + 427.137872746 = 6529.10748341

So rounding to the nearest dollar, it's $6529$? Wait, but maybe the correct answer is $6529$? Wait, no, maybe I made a mistake. Wait, let's check with the formula again. $P=3800$, $r=0.07$, $n=1$, $t=8$. So $A = 3800(1 + 0.07)^8 = 3800*(1.07)^8$. Let's compute $(1.07)^8$:

1.07^1 = 1.07

1.07^2 = 1.07*1.07 = 1.1449

1.07^3 = 1.1449*1.07 = 1.225043

1.07^4 = 1.225043*1.07 = 1.31079601

1.07^5 = 1.31079601*1.07 = 1.4025517307

1.07^6 = 1.4025517307*1.07 = 1.500730351849

1.07^7 = 1.500730351849*1.07 = 1.60578147647843

1.07^8 = 1.60578147647843*1.07 = 1.71818617983192

Then $3800*1.71818617983192 = 3800*1.71818617983192$

Calculate 3800*1.71818617983192:

3800 * 1 = 3800

3800 * 0.7 = 2660

3800 * 0.01818617983192 = 3800 * 0.01 = 38; 3800 * 0.00818617983192 = 31.1074833613

So 38 + 31.1074833613 = 69.1074833613

Then total is 3800 + 2660 = 6460; 6460 + 69.1074833613 = 6529.1074833613

So rounding to the nearest dollar, it's $6529$. Wait, but maybe the system expects a different calculation? Wait, no, maybe I made a mistake in the exponent. Wait, the formula is correct. So the answer should be $6529$.

Answer:

$\$6529$