This question involves the continuous compound interest formula, which is given by:

$A = P e^{rt}$

Where:

• $A$: the balance (amount after interest),
• $P$: the principal (initial deposit),
• $r$: the annual interest rate (in decimal form),
• $t$: time (in years),
• $e$: the mathematical constant (approximately 2.718).

Given:

• $P = 1000$
• $r = 8.5\% = 0.085$
• $t = 5$

We substitute these values into the formula:

$A = 1000 \cdot e^{0.085 \cdot 5}$

Simplification:

$A = 1000 \cdot e^{0.425}$

Let me calculate the exact value of $A$. math

Given values

P = 1000 # Principal r = 0.085 # Interest rate t = 5 # Time in years

Calculate the amount using the formula A = P * e^(rt)

A = P * math.exp(r * t) AThe balance after 5 years, with continuous compounding at an 8.5% annual interest rate, will be approximately $1529.59.

Would you like a detailed breakdown of each step or have further questions?

Here are 5 related questions to expand on this topic:

1. What happens to the balance if the interest rate increases to 10%?
2. How does continuous compounding compare to yearly compounding?
3. What is the balance after 10 years instead of 5 years?
4. How can you rearrange the formula to find the principal $P$ if the balance $A$ is known?
5. What is the effect of doubling the principal on the final amount?

Tip: Continuous compounding always gives the highest return compared to other compounding methods for the same rate and time!