Substitution vs. Tangency Methods for Constrained Optimization

Formulating Zoë's Constrained Optimization Problem

A person has a strict budget of £200 to donate to two charities, Charity A and Charity B. Their personal goal is to choose a donation amount for each charity that results in the greatest possible positive impact, according to their own values. Which of the following scenarios best represents a potential optimal solution to this person's decision-making problem?

Analyzing a Student's Study Plan

A student wants to plan their weekend to maximize their happiness. They have a total of 10 hours for leisure and can choose to spend this time either reading a book or hiking. Match each component of their decision-making problem to its correct description.

The Gardener's Dilemma

A consumer wants to maximize their personal satisfaction by purchasing a combination of two goods, apples and bananas, with a total budget of $10. If the consumer finds a combination of apples and bananas that costs $12 but provides them with the highest possible satisfaction, this combination represents the optimal solution to their problem.

Deconstructing a Production Problem

A company aims to produce the maximum number of widgets possible with a fixed budget of $10,000, which can be spent on two inputs: labor hours and raw materials. Arrange the following steps into the correct logical sequence for solving this problem.

Evaluating an Advertising Strategy

When an individual makes a decision to achieve the best possible outcome, such as maximizing their personal satisfaction, while being limited by a factor like a fixed budget, this limiting factor is known as the ____.

A city planner is tasked with reducing the average commute time for residents. They have a fixed budget of $10 million that can be allocated between two projects: adding new bus routes or increasing the frequency of subway trains. To make the best decision, the planner needs to frame this as a problem of achieving a specific goal under a limitation. Which of the following statements correctly identifies the goal (objective), the decisions to be made (choices), and the limitation (constraint)?