Let's break down the problem step by step.

Exponential Growth and Initial Conditions:

1. Initial Cases on June 1: There were 1170 new positive cases reported.
2. Cases on June 2: There were 1309 new positive cases reported.

Step 1: Finding the Growth Factor

The first task is to find the growth factor that multiplies the number of cases from one day to the next.

• The growth factor can be found by dividing the number of cases on June 2 by the number of cases on June 1:

$\text{Growth Factor} = \frac{\text{Cases on June 2}}{\text{Cases on June 1}} = \frac{1309}{1170} \approx 1.1197$

This means that the number of new cases increased by a factor of approximately 1.1197 from June 1 to June 2.

Step 2: Determine the Exponential Function

An exponential function to model this situation would have the form:

$g(t) = g(0) \cdot r^t$

where:

• $g(0)$ is the initial number of cases (on June 1, $t = 0$).
• $r$ is the growth factor we calculated above.
• $t$ is the number of days since June 1.

Thus, the function $g(t)$ can be written as:

$g(t) = 1170 \cdot (1.1197)^t$

Step 3: Predicting the Number of New Cases

Using the exponential function, we can now predict the number of new cases for the following days.

• Day 0 (June 1): $g(0) = 1170$
• Day 1 (June 2): $g(1) = 1170 \cdot (1.1197) \approx 1309$ (which matches the given data)
• Day 2 (June 3): $g(2) = 1170 \cdot (1.1197)^2 \approx 1464$
• Day 3 (June 4): $g(3) = 1170 \cdot (1.1197)^3 \approx 1636$
• Day 4 (June 5): $g(4) = 1170 \cdot (1.1197)^4 \approx 1828$

Now, let’s address the specific expressions that represent the factor we multiply by:

Step 4: Identifying the Correct Expression

The correct expression that represents the growth factor between any two consecutive days is:

$\frac{f(t+1)}{f(t)}$

For the days given in the problem, the correct expression is:

$\frac{f(1)}{f(0)} = \frac{1309}{1170}$

This is the correct representation of the growth factor.

Conclusion:

• Growth Factor: $1.1197$
• Exponential Function: $g(t) = 1170 \cdot (1.1197)^t$

If you want to predict the number of cases for future days, you can substitute the value of $t$ into the function $g(t)$.

Follow-Up Questions:

1. How would you calculate the number of cases on June 6, 2020?
2. What happens to the number of cases if the growth factor remains constant for 10 days?
3. How does an exponential model differ from a linear model in this context?
4. What assumptions are we making by using an exponential model?
5. Can the growth factor change over time? How would that affect the predictions?
6. How would you adjust the model if new data showed a decrease in the growth factor?
7. What is the percentage increase in cases from June 1 to June 2?
8. How would this model change if the initial number of cases was 1500 instead of 1170?

Tip: Always check the validity of the growth factor over several days to ensure that the exponential model accurately reflects the actual data trend.