Math  /  Algebra

QuestionSuppose that $2000\$ 2000 is invested at a rate of 5.2%5.2 \%, compounded quarterly. Assuming that no withdrawals are made, find the total amount after 5 years.
Do not round any intermediate computations, and round your answer to the nearest cent.

Studdy Solution

STEP 1

1. The principal amount invested is \$2000.
2. The annual interest rate is 5.2%.
3. The interest is compounded quarterly.
4. The investment period is 5 years.
5. We need to find the total amount after 5 years, rounded to the nearest cent.

STEP 2

1. Identify the formula for compound interest.
2. Define the variables in the formula.
3. Substitute the known values into the formula.
4. Compute the total amount.

STEP 3

Identify the formula for compound interest. The formula for compound interest is:
A=P(1+rn)nt A = P \left(1 + \frac{r}{n}\right)^{nt}
where: - A A is the total amount after time t t , - P P is the principal amount, - r r is the annual interest rate (in decimal), - n n is the number of times the interest is compounded per year, - t t is the time in years.

STEP 4

Define the variables in the formula based on the problem statement.
- P=2000 P = 2000 - r=5.2%=0.052 r = 5.2\% = 0.052 - n=4 n = 4 (since the interest is compounded quarterly) - t=5 t = 5

STEP 5

Substitute the known values into the formula:
A=2000(1+0.0524)4×5 A = 2000 \left(1 + \frac{0.052}{4}\right)^{4 \times 5}

STEP 6

Compute the total amount. First, calculate the expression inside the parentheses:
1+0.0524=1+0.013=1.013 1 + \frac{0.052}{4} = 1 + 0.013 = 1.013
Next, calculate the exponent:
4×5=20 4 \times 5 = 20
Now, raise the base to the power of the exponent:
1.013201.296856 1.013^{20} \approx 1.296856
Finally, multiply by the principal amount:
A=2000×1.2968562593.71 A = 2000 \times 1.296856 \approx 2593.71
The total amount after 5 years, rounded to the nearest cent, is:
2593.71 \boxed{2593.71}

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