A few weeks into the deadly SARS (Severe Acute Respiratory Syndrome) epidemic in 2003, the number of cases was increasing by about 4% each day.† On April 1, 2003 there were 1,804 cases. Find an exponential model that predicts the number A(t) of people infected t days after April 1, 2003. A(t)= 1804 * (1.04)^t. Use your model to estimate how fast the epidemic was spreading on April 23, 2003. (Round your answer to the nearest whole number of new cases per day.) ____ cases per day

We are given an exponential growth model for the number of people infected by SARS during the 2003 epidemic:

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

where:
- \( A(t) \) represents the number of infected people on day \( t \),
- \( t \) is the number of days after April 1, 2003,
- \( 1.04 \) represents a 4% daily increase in the number of cases.

To estimate how fast the epidemic was spreading on April 23, 2003, we need to calculate the rate of change of \( A(t) \) at \( t = 22 \) (since April 23 is 22 days after April 1).

### Step 1: Differentiate \( A(t) \) with respect to \( t \)
The rate of change of \( A(t) \) is given by the derivative of the function. We differentiate the exponential function:

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

Using the chain rule, we get:

\[
A'(t) = 1804 \times \ln(1.04) \times (1.04)^t
\]

### Step 2: Calculate \( A'(22) \)
Now, substitute \( t = 22 \) into the derivative to estimate how fast the epidemic was spreading on April 23, 2003:

\[
A'(22) = 1804 \times \ln(1.04) \times (1.04)^{22}
\]

First, calculate \( \ln(1.04) \):

\[
\ln(1.04) \approx 0.03922
\]

Now calculate \( (1.04)^{22} \):

\[
(1.04)^{22} \approx 2.372
\]

Substitute these values into the expression for \( A'(22) \):

\[
A'(22) = 1804 \times 0.03922 \times 2.372
\]

Now calculate:

\[
A'(22) \approx 1804 \times 0.03922 \times 2.372 \approx 167.79
\]

### Final Answer:
Rounding to the nearest whole number, the epidemic was spreading at a rate of approximately **168 cases per day** on April 23, 2003.

Would you like further details or have any questions? Here are some related questions to consider:

1. What would be the predicted number of cases on April 23, 2003?
2. How would the model change if the daily increase were 5% instead of 4%?
3. How long would it take for the number of cases to double at a 4% growth rate?
4. Can the exponential model continue indefinitely, or are there limitations?
5. What is the significance of the constant 1.04 in the model?

**Tip:** Exponential growth models are sensitive to even small changes in the growth rate due to compounding effects over time.

A few weeks into the deadly SARS (Severe Acute Respiratory Syndrome) epidemic in 2003, the number of cases was increasing by about 4% each day. On April 1, 2003 there were 1,804 cases. Find an exponential model that predicts the number A(t) of people infected t days after April 1, 2003. A(t)= 1804 * (1.04)^t. Use your model to estimate how fast the epidemic was spreading on April 23, 2003. (Round your answer to the nearest whole number of new cases per day.) ____ cases per day

Explore how to use an exponential growth model A(t) = 1804 * (1.04)^t to predict the number of SARS cases during the 2003 epidemic. Learn how to calculate the rate of spread on April 23, 2003 using differentiation techniques. Ideal for Grades 11-12.

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