https://file.aiquickdraw.com/mathbot/file/172868468902546rjxw52.jpg

To determine how long it will take for an initial amount of $3000 to grow to $4335 in a savings account with an annual interest rate of 4% compounded quarterly, we can use the formula for compound interest:

\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]

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

Given:
- $ P = 3000 $
- $ A = 4335 $
- $ r = 0.04 $
- $ n = 4 $ (compounded quarterly)

We need to solve for $ t $.

First, we plug the given values into the formula:

\[ 4335 = 3000 \left(1 + \frac{0.04}{4}\right)^{4t} \]

Simplify inside the parentheses:

\[ 4335 = 3000 \left(1 + 0.01\right)^{4t} \]
\[ 4335 = 3000 (1.01)^{4t} \]

Divide both sides by 3000 to isolate the exponential expression:

\[ \frac{4335}{3000} = (1.01)^{4t} \]
\[ 1.445 = (1.01)^{4t} \]

Now, take the natural logarithm of both sides to solve for $ t $:

\[ \ln(1.445) = \ln((1.01)^{4t}) \]
\[ \ln(1.445) = 4t \ln(1.01) \]

Divide both sides by $ 4 \ln(1.01) $:

\[ t = \frac{\ln(1.445)}{4 \ln(1.01)} \]

Now, we can compute this value:

\[ \ln(1.445) \approx 0.3677 \]
\[ \ln(1.01) \approx 0.00995 \]
\[ t = \frac{0.3677}{4 \times 0.00995} \]
\[ t \approx \frac{0.3677}{0.0398} \]
\[ t \approx 9.24 \]

Therefore, it will take approximately 9.24 years for the account to grow to $4335.

Would you like more details or have any questions? Here are five related questions for further practice:

1. How long will it take for the same initial amount to grow to $5000 at the same interest rate compounded quarterly?
2. What will be the future value of $2000 invested for 10 years at an annual interest rate of 5%, compounded monthly?
3. If the interest rate was instead compounded annually, how would the time required to reach $4335 change?
4. How much should be initially invested to reach $10000 in 15 years with an annual interest rate of 3%, compounded quarterly?
5. What is the difference in future value if $3000 is compounded quarterly versus compounded continuously at 4% for 10 years?

**Tip:** When dealing with compound interest problems, ensure you correctly identify the compounding period and convert the interest rate accordingly for accurate calculations.

Suppose that $3000 is placed in a savings account at an annual rate of 4%, compounded quarterly. Assuming that no withdrawals are made, how long will it take for the account to grow to $4335? Do not round any intermediate computations, and round your answer to the nearest hundredth.

Find out how long it will take for an initial investment of $3000 to grow to $4335 with a 4% interest rate compounded quarterly. This problem covers key concepts such as compound interest, exponential growth, and logarithmic equations, ideal for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation