To calculate the total number of shrimp born after 90 years, let's break it down step by step.

1. Shrimp's Reproductive Details:

   • Each shrimp produces 5 broods per year.
   • Each brood contains 40 babies.
   • Therefore, one shrimp produces: $5 \times 40 = 200 \text{ babies per year.}$

2. Shrimp's Lifespan and Generational Impact:

   • A shrimp lives for one year, so it contributes to the next generation only once before dying.
   • The number of shrimp in a single generation multiplies exponentially, as every shrimp contributes to the next generation.

3. Initial Assumption:

   • Starting with 1 shrimp, after 1 year, this shrimp produces 200 offspring.
   • These 200 offspring will, in turn, each produce 200 shrimp the following year.
   • This process continues for 90 years.

4. Calculating Total Shrimp Born: The growth is exponential, represented by: $N = \text{Initial shrimp count} \times (\text{Number of offspring per shrimp})^t,$ where $t$ is the number of generations (90 years).

   Here:

   • Initial shrimp count = 1.
   • Number of offspring per shrimp = 200.
   • $t = 90$.

   Substituting: $N = 1 \times 200^{90}.$

   This results in a very large number, as $200^{90}$ grows astronomically due to the exponential nature of the equation. A more realistic model would likely consider environmental constraints like food, space, and predation. Would you like to see this enormous number calculated explicitly?

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Questions for Deeper Understanding:

1. Would you like to explore the mathematical growth of populations in real-world scenarios with constraints?
2. Shall I calculate $200^{90}$ explicitly for you?
3. Should we consider a model that includes factors like shrimp mortality rates or carrying capacity?
4. Would you like an explanation of exponential growth versus logistic growth?
5. Do you need help plotting population growth over time for a better visual understanding?

Tip: When calculating exponential growth, be mindful of real-world factors that can limit the population.