The problem asks for the growth rate of the population at time $t = 50$, based on the graph provided. The graph shows the development of the population $N$ of a particular animal species as a function of time $t$ (in days).

To find the growth rate at $t = 50$, we need to calculate the slope of the graph at that point, which represents the derivative $\frac{dN}{dt}$ at $t = 50$.

Approach:

1. Visual inspection: From the graph, at $t = 50$, the population seems to have stabilized or slowed down significantly.
2. Estimate the slope: The slope can be approximated as the change in $N$ over a small interval of time around $t = 50$. We can approximate this by finding two nearby points on the curve and calculating the slope between them.

Let's go through the details:

• At $t = 40$, $N \approx 95$ individuals.
• At $t = 60$, $N \approx 105$ individuals.

Thus, over the interval from $t = 40$ to $t = 60$: $\frac{\Delta N}{\Delta t} = \frac{105 - 95}{60 - 40} = \frac{10}{20} = 0.5 \, \text{individuals per day}.$

This provides an approximation for the growth rate at $t = 50$, which is about $0.5$ individuals per day.

Let me know if you'd like more detailed information or further analysis!

Related Questions:

1. What would the growth rate be at $t = 30$ based on the graph?
2. How can we mathematically model this type of population growth?
3. What happens to the population growth rate as $t$ increases further beyond $t = 100$?
4. How does the initial growth rate compare with the growth rate at $t = 50$?
5. Can you find the inflection point of the graph, where the growth rate changes from increasing to decreasing?

Tip:

For precise calculation, you can use calculus by deriving the equation of the curve if it's available or use numerical methods to estimate the derivative at specific points.