Let’s break this problem into its parts:

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Part (a): Exponential Formula for the Balance

The general formula for continuous compounding is:

$A(t) = P e^{rt}$

Where:

• $A(t)$: The account balance at time $t$,
• $P = 250$: The initial investment,
• $r = 0.0575$: The annual percentage rate (APR) as a decimal,
• $t$: Time in years.

Substitute the values into the formula:

$A(t) = 250 e^{0.0575t}$

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Alternative Form

An exponential function can also be written in the standard base-2 form:

$A(t) = A_0 \cdot (b)^{t}$

Here:

• $b = e^r = e^{0.0575} \approx 1.0592$,
• $A_0 = 250$.

Thus:

$A(t) = 250 \cdot (1.0592)^t$

Summary of the Two Forms:

1. Standard (Base $e$) Form: $A(t) = 250 e^{0.0575t}$
2. Alternative (Base 10) Form: $A(t) = 250 \cdot (1.0592)^t$

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Part (b): Account Balance After 5 Years

We will calculate the account balance using both forms:

1. Using Standard Form: $A(5) = 250 e^{0.0575 \cdot 5} = 250 e^{0.2875} \approx 250 \cdot 1.333 \approx 333.25$

2. Using Alternative Form: $A(5) = 250 \cdot (1.0592)^5 \approx 250 \cdot 1.333 \approx 333.25$

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Which is More Accurate?

Both forms yield the same result because they are mathematically equivalent. However:

• The standard form $A(t) = 250 e^{0.0575t}$ is often considered more precise since it uses the natural exponential base $e$ directly, which avoids rounding errors inherent in using approximations for $e^{0.0575}$ like $1.0592$.

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Final Answers

(a) The formulas are:

• Standard Form: $A(t) = 250 e^{0.0575t}$
• Alternative Form: $A(t) = 250 \cdot (1.0592)^t$

(b) Account Balance After 5 Years:

• Using Standard Form: $\$333.25$
• Using Alternative Form: $\$333.25$

Both forms provide the same result, but the standard form is generally more accurate for theoretical purposes.

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Do you have any questions or need additional explanations?
Here are some follow-up questions to explore this concept further:

1. How would the formula change if the APR were compounded annually instead of continuously?
2. What happens to the account balance if the APR increases to 6%?
3. How long would it take for the balance to double with continuous compounding?
4. Can you rewrite the exponential function using logarithms?
5. What is the significance of the constant $e$ in financial calculations?

Tip: Continuous compounding maximizes the growth of your investment because it assumes the interest is being added infinitely often over time!