Ashley F.
asked • 04/30/23pre calc question
A savings account pays interest at a annual percentage rate (APR) of 1.7%, compounded quarterly.
A customer opens a savings account with an initial deposit of $3000. Assuming no further deposits or withdrawals, after how many years will the value of the account reach $3700?
Round your answer to three decimal places.
Answer: The value will reach $3700 after years.
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1 Expert Answer
3700 = 3000(1+.017/4)^4t
37/30 = 1.00425^4t
4t = log1.00425(37/30) = ln(37/30)/ln1.00425 = .20972/.00424099 =49.45
t= 12.363 years
= 12 years and 132 days
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Juan M.
To solve this problem, we'll use the formula for compound interest: A = P(1 + r/n)^(nt) where A is the final amount, P is the principal (initial deposit), r is the annual interest rate (as a decimal), n is the number of times interest is compounded per year, and t is the number of years. We are given the following information: - P = $3000 (initial deposit) - r = 0.017 (1.7% as a decimal) - n = 4 (compounded quarterly) - A = $3700 (final amount) We want to find t (the number of years). Plugging in the values and solving for t: 3700 = 3000(1 + 0.017/4)^(4t) To find t, we'll follow these steps: 1. Divide both sides of the equation by 3000: 3700/3000 = (1 + 0.017/4)^(4t) 2. Simplify the left side of the equation: 1.2333 ≈ (1 + 0.017/4)^(4t) 3. Take the natural logarithm (ln) of both sides of the equation: ln(1.2333) = ln((1 + 0.017/4)^(4t)) 4. Use the logarithm power rule to bring the exponent to the front: ln(1.2333) = 4t * ln(1 + 0.017/4) 5. Solve for t by dividing both sides by (4 * ln(1 + 0.017/4)): t ≈ ln(1.2333) / (4 * ln(1 + 0.017/4)) 6. Plug the values into a calculator to find t: t ≈ ln(1.2333) / (4 * ln(1.00425)) 7. Calculate the result: t ≈ 6.351 years Rounding the answer to three decimal places, it will take approximately 6.351 years for the value of the account to reach $3700.04/30/23