Optimization II – Part 1

Optimization Problems in Calculus

Definition of Optimization Problem

  • An optimization problem in calculus seeks to find the maximum or minimum value of a given function, typically subject to certain constraints.

Example Scenario

Example 1

  • A farmer has 60 feet of fencing to construct a rectangular garden, which includes an additional fence that divides the garden into two equal parts.

  • Objective: Find the garden dimensions that yield the maximum area.

Characteristic Features of Optimization Problems

  1. The Objective Quantity:

    • This is the quantity we wish to maximize or minimize.

  2. The Constraints:

    • Conditions that limit the values the objective quantity can take.

  3. The Variables:

    • The quantities that can be controlled or adjusted to affect the objective quantity.

  4. The Final Answer Format:

    • Indicates which quantities are required in the final answer.

  5. Additional Numerical Information:

    • Numerical data that assists in establishing mathematical expressions for the objective quantity and constraints.

Steps for Solving Optimization Problems

Example 1 Solution Steps

  1. Variable Assignment:

    • Assign letters to represent each variable in the problem (e.g. Length = L, Width = W).

    • If applicable, draw and label a diagram to visualize the problem.

  2. Expressing the Objective Quantity:

    • The objective quantity to optimize in this instance is the area of the rectangular garden, expressed as:
      Area=LimesWArea = L imes W

  3. Using Constraints to Formulate Objective Function:

    • Utilize the constraint given by the total length of fencing.

    • The fencing constraints can be written as:
      2L+3W=602L + 3W = 60

    • Solve this equation for one variable (either L or W) to express the area solely as a function of one variable.

  4. Finding the Domain:

    • Identify the domain of the function based on physical restrictions: both length (L) and width (W) must be greater than or equal to 0.

  5. Optimizing the Function:

    • Optimize the area function over its domain using methods outlined in Section 4.4.

    • Consider whether the domain is closed or open:

      • Closed Interval: Calculate function values at critical points and endpoints.

      • Open Interval: Use graphical methods or other techniques.

  6. Revisiting the Problem Statement:

    • Carefully read back over the problem and ensure that the solution includes appropriate units.

Alternative Implementation of Steps 4 and 5

  • In Step 4, it was established that both L ≥ 0 and W ≥ 0, allowing for the possibility that they could be zero.

  • If both L and W are zero, the area becomes zero, yielding an unrealistic garden scenario.

  • Deciding whether to accept L and W being zero affects the classification of the domain (closed vs open), thereby influencing the calculus techniques applicable in Step 5.

Guidelines for Solving Optimization Problems

  1. Assign letters to each variable mentioned in the problem; often the prompt provides clues for this.

  2. Derive an expression for the objective quantity needing optimization.

  3. Use the provided constraint to express the objective as a function of a single variable.

  4. Establish the domain of the function based upon the problem's physical constraints.

  5. Perform optimization on the function utilizing methods from Section 4.4, keeping in mind the nature of the domain.

    • Closed Interval: Assess values at critical points and bounds.

    • Open Interval: Employ graphical or alternative methods.

  6. Carefully review the prompt to finalize the answer, ensuring units are included.

Supplemental Examples of Optimization Problems

Example 2

  • Scenario: Building a sandcastle on a beachfront with one side parallel to the shore, where the side facing the shore requires stronger construction.

  • Construction Time Required:

    • 40 minutes per meter for the strong side

    • 10 minutes per meter for each of the weaker sides

  • Total time available for construction: 180 minutes

    • Questions:

      • What dimensions (length and width) maximize the castle's area?

      • What is the area of the maximized castle?

Example 3

  • Scenario: Constructing an open box from an 8-inch square tin sheet by cutting identical squares from each corner and folding up the resulting flaps.

  • Question: What dimensions yield the largest possible volume for the box?

MATH 221 University of Delaware