https://file.aiquickdraw.com/mathbot/file/17260847492614aewrk2g.jpg

In the image you uploaded, there is a logistic function describing the zombie population growth over time, which is modeled by the function:

\[
Z(t) = \frac{315}{1 + 14e^{-0.23t}}
\]

Let's break down the tasks and calculations:

### 1. Complete the Table
To complete the table for the values of \( t = 0, 10, 20, 30, 40, 50, 60, \) and \( 70 \) days, we will compute \( Z(t) \) for each time point.

### 2. Find the Limiting Value of the Function
The limiting value of this logistic function as \( t \to \infty \) is the carrying capacity, which is the numerator of the function, 315. This is the maximum expected number of zombies in thousands.

### 3. Concavity of the Function
To determine when the function is concave up, we need to find the second derivative of the function and find when it is positive.

Let's start by calculating the zombie population values for the table and then explore the concavity of the function.Here are the calculated zombie populations (in thousands) for each time point:

- \( t = 0 \): 21,000 zombies
- \( t = 10 \): 131,052 zombies
- \( t = 20 \): 276,140 zombies
- \( t = 30 \): 310,618 zombies
- \( t = 40 \): 314,555 zombies
- \( t = 50 \): 314,955 zombies
- \( t = 60 \): 314,996 zombies
- \( t = 70 \): 314,999 zombies

Next, let's determine the intervals of \( t \) for which the function \( Z(t) \) is concave up by finding and analyzing the second derivative.It seems like I can’t do more advanced data analysis right now. Please try again later.

If you have other questions or need clarification on what we've covered so far, feel free to ask! Here are a few more things you might be interested in:

1. How to graph the function \(Z(t)\) manually or on a platform like Desmos?
2. More details on how logistic functions like \(Z(t)\) model population growth.
3. How to find the inflection point of a function mathematically.
4. Further exploration into concavity and convexity in functions.
5. Any specific part of the logistic growth model you'd like to understand better?

**Tip**: When working with logistic growth functions, always check the behavior as \( t \to \infty \) and \( t \to -\infty \) to fully understand the limits and long-term trends of the model.

Keeping with the same zombie theme, it is estimated that the nuclear fallout from the Tahlequah Nuclear Power Plant, as estimated by George Romero, will lead to a zombie population that follows a logistic growth model described by the function Z(t) = 315 / (1 + 14e^-0.23t), where Z is in thousands and t is measured in days. Tasks include graphing the function, determining the limiting value, and estimating the t-values for concavity changes.

Analyze a zombie population logistic growth function with Z(t) = 315 / (1 + 14e^-0.23t), determining the limiting value of 315 and finding the t-values for concave-up behavior. Suitable for students learning advanced calculus.

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