We are given the formula $A = 10,000 e^{0.01t}$, and we need to evaluate it for $t = 3$, $t = 7$, and $t = 9$, rounding to the nearest cent.

Let's compute each case:

1. For $t = 3$:

$A = 10,000 e^{0.01 \times 3} = 10,000 e^{0.03}$

2. For $t = 7$:

$A = 10,000 e^{0.01 \times 7} = 10,000 e^{0.07}$

3. For $t = 9$:

$A = 10,000 e^{0.01 \times 9} = 10,000 e^{0.09}$

Let me perform these calculations.

math

Given values

t_values = [3, 7, 9] base_amount = 10000

Calculate A for each t

A_values = [base_amount * math.exp(0.01 * t) for t in t_values]

Round results to nearest cent

A_values_rounded = [round(A, 2) for A in A_values]

A_values_roundedThe values of $A$ rounded to the nearest cent for $t = 3$, $t = 7$, and $t = 9$ are:

• For $t = 3$: $A = 10,304.55$
• For $t = 7$: $A = 10,725.08$
• For $t = 9$: $A = 10,941.74$

Would you like further details on these calculations, or have any other questions?

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Here are five related questions you can explore next:

1. How does the value of $t$ affect the growth of $A$?
2. What is the concept of exponential growth and how is it used in real-world applications?
3. How would the value of $A$ change if the exponent was increased to $0.05t$ instead of $0.01t$?
4. Can you explain the importance of the constant $e$ in this equation?
5. How would the calculations change if $A$ started with a different initial value, say $5,000$?

Tip: In exponential growth models, even small changes in the exponent can have large effects over time!