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Math  /  Algebra

QuestionKala-needs $8346\$ 8346 for a future project. She can invest $6000\$ 6000 now at an annual rate of 3.4%3.4 \%, compounded quarterly. Assuming that no withdrawals are made, how long will it take for her to have enough money for her project?
Do not round any intermediate computations, and round your answer to the nearest hundredth. □\square years

Studdy Solution

STEP 1

1. Kala needs \$8346 for her future project.
2. She can invest \$6000 now.
3. The annual interest rate is 3.4%, compounded quarterly.
4. No withdrawals are made during the investment period.
5. We need to find out how long it will take for the investment to grow to \$8346.

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. Solve for the time period required.

STEP 3

Identify the formula for compound interest. The formula 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 (the initial amount of money). - 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

Define the variables in the formula based on the problem:
- A=8346 A = 8346 - P=6000 P = 6000 - r=0.034 r = 0.034 (3.4% as a decimal) - n=4 n = 4 (compounded quarterly)

STEP 5

Substitute the known values into the compound interest formula:
8346=6000(1+0.0344)4t 8346 = 6000 \left(1 + \frac{0.034}{4}\right)^{4t}

STEP 6

Solve for the time period t t .
First, simplify the expression inside the parentheses:
1+0.0344=1+0.0085=1.0085 1 + \frac{0.034}{4} = 1 + 0.0085 = 1.0085
Substitute back into the equation:
8346=6000×(1.0085)4t 8346 = 6000 \times (1.0085)^{4t}
Divide both sides by 6000 to isolate the exponential term:
83466000=(1.0085)4t \frac{8346}{6000} = (1.0085)^{4t}
1.391=(1.0085)4t 1.391 = (1.0085)^{4t}
Take the natural logarithm of both sides to solve for t t :
ln⁡(1.391)=ln⁡((1.0085)4t) \ln(1.391) = \ln((1.0085)^{4t})
Using the logarithmic identity ln⁡(ab)=bln⁡(a) \ln(a^b) = b \ln(a) , we have:
ln⁡(1.391)=4t⋅ln⁡(1.0085) \ln(1.391) = 4t \cdot \ln(1.0085)
Solve for t t :
t=ln⁡(1.391)4⋅ln⁡(1.0085) t = \frac{\ln(1.391)}{4 \cdot \ln(1.0085)}
Calculate the value:
t≈0.32934×0.00847 t \approx \frac{0.3293}{4 \times 0.00847}
t≈0.32930.03388 t \approx \frac{0.3293}{0.03388}
t≈9.72 t \approx 9.72
Kala will need approximately 9.72 \boxed{9.72} years for her investment to grow to \$8346.

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