Math Problem Statement
Second National Bank offers an account that earns interest at an annual rate of 7.1%, com-
pounded continuously. You take advantage of this great interest rate and deposit money into
this account. Your account grows to $4850.45 in three years.
(a) How much did you initially deposit into this account?
(Round your answer to the nearest cent.)
(b) Suppose you want to go on a trip in three years and you need your deposit to grow to $6000,
instead of $4850.45. If you can only afford to deposit $4000 in the account today, what
interest rate do you need the bank to offer instead, so that you to have the $6000 in three
years? (Round your answer to two decimal places, after writing your answer as
a percentage.)
Solution
(a) How much did you initially deposit into this account?
The formula for continuous compounding is:
where:
- is the final amount in the account.
- is the principal (initial deposit).
- is the interest rate (in decimal form).
- is the time in years.
Given:
- (7.1% annual rate)
- years
We need to find the initial deposit, .
Rearranging the formula to solve for :
Substitute the given values:
Calculating the exponent:
Now, calculate and then divide by that result.
So,
Answer for (a): The initial deposit was approximately $3920.00.
(b) What interest rate do you need for $4000 to grow to $6000 in three years?
Now we need to find the interest rate if:
- The initial deposit
- The final amount
- The time years
Using the continuous compounding formula:
Rearrange to solve for :
Substitute the values:
Calculate the fraction inside the logarithm:
So,
Calculate :
Convert to a percentage by multiplying by 100:
Answer for (b): You would need an interest rate of approximately 13.52% for your deposit to grow to $6000 in three years.
Would you like more detailed steps or further clarification on any part of this solution?
Here are some additional related questions:
- How would the final amount change if the compounding was quarterly instead of continuous?
- What would the initial deposit need to be for an account to grow to $7000 in five years at a 7.1% continuous interest rate?
- If the interest rate changes after the first year, how would we adjust the formula to calculate the final amount?
- What happens to the growth of the account if the compounding frequency is daily instead of continuously?
- How long would it take for $4000 to double with a 7.1% continuous interest rate?
Tip: Continuous compounding grows faster than other compounding frequencies due to the mathematical properties of , Euler's number.
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Math Problem Analysis
Mathematical Concepts
Exponential Growth
Continuous Compounding
Logarithms
Interest Rate Calculations
Formulas
A = P * e^(rt)
r = (ln(A/P)) / t
Theorems
Exponential Growth
Properties of Logarithms
Suitable Grade Level
Grades 11-12
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