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We are given an exponential model for the number of people infected with SARS after April 1, 2003:

\[
A(t) = 1804 \cdot (1.04)^t
\]

where \( A(t) \) represents the number of cases after \( t \) days, and the rate of increase is 4% per day.

To estimate how fast the epidemic was spreading on April 28, 2003, we need to calculate the rate of change of \( A(t) \), i.e., we need the derivative of \( A(t) \), and then evaluate it at \( t = 27 \) (since April 28 is 27 days after April 1).

### Step 1: Differentiate \( A(t) \)
Using the chain rule, we find the derivative of \( A(t) \):

\[
A'(t) = \frac{d}{dt} \left( 1804 \cdot (1.04)^t \right)
\]
\[
A'(t) = 1804 \cdot \ln(1.04) \cdot (1.04)^t
\]

### Step 2: Plug in \( t = 27 \)
Now, substitute \( t = 27 \) into the derivative to find the rate of change at that time:

\[
A'(27) = 1804 \cdot \ln(1.04) \cdot (1.04)^{27}
\]

We can now compute this:

\[
A'(27) \approx 1804 \cdot 0.03922 \cdot (1.04)^{27}
\]
\[
(1.04)^{27} \approx 2.9257
\]
\[
A'(27) \approx 1804 \cdot 0.03922 \cdot 2.9257
\]
\[
A'(27) \approx 206.7 \quad \text{(rounding to the nearest whole number: } 207)
\]

Thus, the epidemic was spreading at approximately **207 new cases per day** on April 28, 2003.

Would you like further details or have any questions about this calculation? Here are some related questions to deepen your understanding:

1. What is the significance of the 4% growth rate in the model?
2. How would the model change if the growth rate were higher or lower?
3. Can you explain the meaning of the derivative in the context of this problem?
4. What happens to the number of cases as \( t \) increases significantly?
5. How could we modify the model if interventions slowed the growth rate?

**Tip:** In exponential models, small changes in the growth rate can have a large impact on long-term outcomes.

Use the exponential model A(t) = 1804 * (1.04)^t to estimate how fast the SARS epidemic was spreading on April 28, 2003. The number of cases on April 1, 2003, was 1,804. Round the answer to the nearest whole number of new cases per day.

Learn how to calculate the rate of change for the SARS epidemic spread using an exponential growth model. Given that the number of cases increased by 4% per day, the solution includes using differentiation to find the spread rate on April 28, 2003. Suitable for students in Grades 10-12.

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