Studdy AI Logo
Math  /  Algebra

QuestionKatherin deposits \$75,000 in a 3-year CD account that yields 3.5\% interest, compounded weekly. What is the amount of his interest, to the nearest cent, at the end of 3 years?

Studdy Solution

STEP 1

1. The principal amount deposited is \$75,000.
2. The interest rate is 3.5% per annum.
3. The interest is compounded weekly.
4. The time period is 3 years.
5. We need to find the total interest earned at the end of 3 years.

STEP 2

1. Identify the formula for compound interest.
2. Determine the values for each variable in the formula.
3. Calculate the compound interest.
4. Calculate the total interest earned.

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 amount of money accumulated after n years, including interest. - P P is the principal amount (initial deposit). - r r is the annual interest rate (decimal). - n n is the number of times that interest is compounded per year. - t t is the time the money is invested for in years.

STEP 4

Determine the values for each variable in the formula.
- P=75000 P = 75000 - r=0.035 r = 0.035 (since 3.5% as a decimal is 0.035) - n=52 n = 52 (since interest is compounded weekly, there are 52 weeks in a year) - t=3 t = 3

STEP 5

Calculate the compound interest using the formula:
A=75000(1+0.03552)52×3 A = 75000 \left(1 + \frac{0.035}{52}\right)^{52 \times 3}
First, calculate 0.03552 \frac{0.035}{52} :
0.03552≈0.0006731 \frac{0.035}{52} \approx 0.0006731
Now calculate 52×3=156 52 \times 3 = 156 .
Substitute these values back into the formula:
A=75000(1+0.0006731)156 A = 75000 \left(1 + 0.0006731\right)^{156}
Calculate the expression inside the parentheses:
1+0.0006731≈1.0006731 1 + 0.0006731 \approx 1.0006731
Raise this to the power of 156:
(1.0006731)156≈1.1096 (1.0006731)^{156} \approx 1.1096
Now calculate A A :
A=75000×1.1096≈83220 A = 75000 \times 1.1096 \approx 83220

STEP 6

Calculate the total interest earned by subtracting the principal from the accumulated amount:
Interest=A−P \text{Interest} = A - P Interest=83220−75000 \text{Interest} = 83220 - 75000 Interest=8220 \text{Interest} = 8220
The amount of interest earned, to the nearest cent, at the end of 3 years is:
8220 \boxed{8220}

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord