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Exponential and Logarithmic Functions
Finding the time in a word problem on compound interest
Morg:
Teresa needs for a future project. She can invest now at an annual rate of , compounded monthly. 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.
years
Studdy Solution
STEP 1
1. Teresa needs a future amount of 4000 now.
3. The annual interest rate is 10.8%, compounded monthly.
4. No withdrawals are made during the investment period.
5. We need to find the time it will take for the investment to grow to $6260.
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 unknown variable, which is time.
STEP 3
Identify the formula for compound interest. The formula is:
where:
- is the future value of the investment/loan, including interest.
- is the principal investment amount (initial deposit or loan amount).
- is the annual interest rate (decimal).
- is the number of times that interest is compounded per year.
- is the number of years the money is invested for.
STEP 4
Define the variables in the formula based on the problem:
-
-
-
- (since the interest is compounded monthly)
STEP 5
Substitute the known values into the formula:
STEP 6
Solve for the unknown variable, .
First, simplify the expression inside the parentheses:
Now, substitute back into the equation:
Divide both sides by 4000 to isolate the exponential term:
Calculate the left side:
Take the natural logarithm of both sides to solve for :
Using the logarithmic identity , we have:
Solve for :
Calculate the value of :
The time it will take for Teresa's investment to grow to $6260 is approximately \( \boxed{4.16} \) years.
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