Math  /  Algebra

QuestionKyoko has \12,000thatshewantstoinvest.Herbankhasseveralinvestmentaccountstochoosefrom,allcompoundingdaily.Hergoalistohave12,000 that she wants to invest. Her bank has several investment accounts to choose from, all compounding daily. Her goal is to have \14,000 14,000 by the time she finishes graduate school in 6 years. To the nearest hundredth of a percent, what should her minimum annual interest rate be in order to reach her goal? (Hint : solve the compound interest formula for the interest rate.)
The minimum annual interest rate is \square \%.
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Studdy Solution

STEP 1

1. Kyoko has an initial investment of \$12,000.
2. She wants to have \$14,000 after 6 years.
3. The interest is compounded daily.
4. We need to find the minimum annual interest rate, compounded daily, to achieve this goal.

STEP 2

1. Understand the compound interest formula.
2. Set up the equation using the given values.
3. Solve the equation for the interest rate.
4. Convert the interest rate to a percentage and round to the nearest hundredth.

STEP 3

Understand the compound interest formula. 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 investment). - 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

Set up the equation using the given values. Here, A=14000 A = 14000 , P=12000 P = 12000 , n=365 n = 365 (since the interest is compounded daily), and t=6 t = 6 .
14000=12000(1+r365)365×6 14000 = 12000 \left(1 + \frac{r}{365}\right)^{365 \times 6}

STEP 5

Solve the equation for the interest rate r r .
First, divide both sides by 12000:
1400012000=(1+r365)2190 \frac{14000}{12000} = \left(1 + \frac{r}{365}\right)^{2190}
Simplify the left side:
1.1667=(1+r365)2190 1.1667 = \left(1 + \frac{r}{365}\right)^{2190}
Take the 2190th root of both sides to isolate the term with r r :
(1.1667)12190=1+r365 \left(1.1667\right)^{\frac{1}{2190}} = 1 + \frac{r}{365}
Subtract 1 from both sides:
(1.1667)121901=r365 \left(1.1667\right)^{\frac{1}{2190}} - 1 = \frac{r}{365}
Multiply both sides by 365 to solve for r r :
r=365×((1.1667)121901) r = 365 \times \left(\left(1.1667\right)^{\frac{1}{2190}} - 1\right)

STEP 6

Convert the interest rate to a percentage and round to the nearest hundredth.
Calculate r r using a calculator:
r0.0271 r \approx 0.0271
Convert to a percentage:
r2.71% r \approx 2.71\%
The minimum annual interest rate is:
2.71% \boxed{2.71\%}

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