Exploration 4.1.1.
The witch Agnesi is brewing a healing elixir and a poison. A pint of healing elixir takes 3 newt eyes and one frog, whereas a pint of poison takes 1 each of newt eyes and frogs. She currently has 34 newt eyes and 14 frogs.
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The name Agnesi was chosen by my Spring 2024 class who knew her from her eponymous curve.
Supposing that the healing elixir sells for three gold pieces, and the poison sells for two. Agnesi wishes to maximize her revenue. Let us suppose that since these are liquids, she is happy making fractional amounts of elixirs and potions.
(a)
Before proceeding to solve the problem, make an estimate: how much do you think each newt eye and frog is worth to her? Why do you think so?
(b)
We now return to the initially posed maximization problem. Sketch the feasible region for this problem, and use whatever method you feel like to find the optimal solution.
(c)
Agnesi is frustrated by her production levels and income. She is going to recruit some local children to gather more materials for her. Without explicitly computing anything, looking at her situation, what would result in greater profits for her, more newt eyes or frogs?
(d)
Recompute this linear optimization problem with 35 newt eyes and 14 frogs, then with 34 newt eyes and 15 frogs. Which provides the greater increase in revenue? Is this consistent with what you thought earlier?
(e)
If the need for healing elixir increases and they now sell for 5 gold, would that change our answers above?