Question: An entomologist studying pollination patterns models the number of flowers visited by bees as $ f(t) = -5t^2 + 30t + 100 $. What is the maximum number of flowers the bees visit in a day? - Imagemakers

Understanding the Context

Here, $ t $ represents time in hours, and $ f(t) $ represents the number of flowers visited at time $ t $. But what does this model truly reveal about bee behavior? And crucially, what is the maximum number of flowers bees visit in a single day according to this model?

The Mathematical Model Explained

The function $ f(t) = -5t^2 + 30t + 100 $ is a quadratic equation, and its graph forms a parabola that opens downward because the coefficient of $ t^2 $ is negative ($ -5 $). This means the function has a peak — a maximum point — which occurs at the vertex of the parabola.

Key Insights

For a quadratic function in the form $ f(t) = at^2 + bt + c $, the time $ t $ at which the maximum (or minimum) occurs is given by:
$$ t = -rac{b}{2a} $$

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

So, the maximum number of flowers visited happens 3 hours after tracking begins.

Calculating the Maximum Number of Flowers Visited

Final Thoughts

Now, substitute $ t = 3 $ back into the original function to find the maximum number of flowers visited:
$$ f(3) = -5(3)^2 + 30(3) + 100 $$
$$ f(3) = -5(9) + 90 + 100 $$
$$ f(3) = -45 + 90 + 100 $$
$$ f(3) = 145 $$

Therefore, the maximum number of flowers the bees visit in a day, according to this model, is 145 flowers.

Practical Implications for Pollination Research

This mathematical insight helps entomologists and ecologists understand the peak foraging period, guiding research on:

• Optimal timing for bee conservation efforts
  
• Understanding environmental factors affecting pollinator activity (such as temperature, floral abundance, and pesticide exposure)
  
• Maximizing pollination efficiency in agricultural fields