Solution: The function $ f(t) = -5t^2 + 30t + 100 $ is a quadratic opening downward. The vertex occurs at $ t = \frac-b2a = \frac-302(-5) = 3 $. Substituting $ t = 3 $, $ f(3) = -5(9) + 30(3) + 100 = -45 + 90 + 100 = 145 $. The maximum number of flowers visited is $\boxed145$. - Coaching Toolbox

Understanding the Context

The function $ f(t) = -5t^2 + 30t + 100 $ is a classic example of a quadratic equation in the standard form $ f(t) = at^2 + bt + c $, where:

• $ a = -5 $,
  
• $ b = 30 $,
  
• $ c = 100 $.

Since the coefficient $ a $ is negative ($ -5 < 0 $), the parabola opens downward, meaning it has a single maximum point (vertex) rather than a minimum. This downward curvature perfectly mirrors real-world phenomena where growth or visitation initially rises and then declines—such as peak pollinator activity at the optimal time.

Finding the Time of Maximum Visitation

The vertex of a parabola defined by $ f(t) = at^2 + bt + c $ occurs at $ t = rac{-b}{2a} $. Substituting the values:

Key Insights

$$
t = rac{-b}{2a} = rac{-30}{2(-5)} = rac{-30}{-10} = 3
$$

Thus, the maximum number of flower visits occurs at $ t = 3 $ hours after observation begins—ideal for coordinating conservation efforts or monitoring pollinator behavior.

Calculating the Maximum Value

To find how many flowers are visited at this peak, substitute $ t = 3 $ back into the original function:

$$
f(3) = -5(3)^2 + 30(3) + 100 = -5(9) + 90 + 100 = -45 + 90 + 100 = 145
$$

Final Thoughts

This means the maximum number of flower visits reaches 145, highlighting peak pollinator activity at the optimal hour.

Why This Matters in Ecology

By modeling pollinator behavior with such quadratic functions, scientists and conservationists gain actionable insights. The precise calculation of $ f(3) = 145 $ enables:

• Timed interventions to protect vulnerable plant species,
  
• Predictive analytics for pollination efficiency,
  
• Better understanding of how environmental changes affect floral attraction and visitation.

Conclusion

The quadratic function $ f(t) = -5t^2 + 30t + 100 $ exemplifies how mathematics enhances ecological research. With a klar logically derived maximum of $ oxed{145} $ flower visits at $ t = 3 $, this model serves as a vital tool in safeguarding pollinator health and supporting biodiversity.

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