The image shows a mathematical function related to a logistic growth model for confirmed cases of a virus in a town named "Maths Town." The function $C(t)$ is given as follows:

$C(t) = \frac{1000}{1 + e^{5(t - 10)}}$

where $C(t)$ represents the cumulative number of confirmed cases after $t$ days, with $t \geq 0$.

The function has the typical form of a logistic function, which models growth that initially increases rapidly, then slows down as it approaches an upper limit or carrying capacity. In this case:

• The carrying capacity, or maximum number of cases, is 1000.
• The growth rate is affected by the exponent $5(t - 10)$, which indicates that at $t = 10$, the growth rate is changing significantly, possibly reaching the midpoint of the logistic curve.

It appears the problem might involve analyzing or proving properties of this function, possibly by showing how it behaves as $t \to \infty$ or by calculating specific values or derivatives.

Would you like further details on any specific aspect of this function? Here are some related questions for deeper exploration:

1. What are the initial conditions for $C(t)$ when $t = 0$?
2. What is the behavior of $C(t)$ as $t \to \infty$?
3. How does the value $t = 10$ relate to the growth of $C(t)$, and why is it significant?
4. What is the rate of change $C'(t)$ of confirmed cases at any time $t$?
5. How does this logistic model compare to an exponential model in terms of case growth?

Tip: Logistic functions are widely used to model population growth, where there is an initial exponential growth phase, followed by a slowdown as the population approaches a limit due to resource constraints.