To calculate the average price of a two-bedroom apartment in 2023, we need to first determine the value of $t$ corresponding to the year 2023. Since $t = 0$ represents 1994:

$t = 2023 - 1994 = 29$

The price function is given as:

$p(t) = 0.33e^{0.16t}$

Step 1: Calculate $p(29)$

Substitute $t = 29$ into the equation:

$p(29) = 0.33e^{0.16 \cdot 29}$

Evaluate this using a calculator:

$p(29) = 0.33e^{4.64}$

Using $e^{4.64} \approx 103.56$:

$p(29) = 0.33 \cdot 103.56 = 34.17 \, \text{million dollars}$

The average price of a two-bedroom apartment in downtown New York City in 2023 is approximately 34 million dollars.

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Step 2: How fast was the price increasing in 2023?

The rate of change of the price is given by the derivative of $p(t)$:

$p'(t) = \frac{d}{dt} \left( 0.33e^{0.16t} \right) = 0.33 \cdot 0.16e^{0.16t}$

Substitute $t = 29$:

$p'(29) = 0.33 \cdot 0.16 \cdot e^{4.64}$

Using $e^{4.64} \approx 103.56$:

$p'(29) = 0.33 \cdot 0.16 \cdot 103.56 = 5.46 \, \text{million dollars per year}$

The price was increasing at a rate of approximately 5.5 million dollars per year in 2023.

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Final Results:

1. Average price in 2023: $\mathbf{34} \, \text{million dollars}$
2. Rate of increase in 2023: $\mathbf{5.5} \, \text{million dollars per year}$

Would you like further details or explanations?
Here are 5 related questions for exploration:

1. How does the rate of change vary with time $t$?
2. What was the average price in 1994?
3. How would the model change if $t = 0$ represented a different year?
4. What does the exponential growth rate imply about housing markets?
5. How accurate is this model for predictions beyond 2004, given its domain of validity?

Tip: Exponential growth models can rapidly diverge from reality for long-term predictions; always assess their assumptions!