Building a Model

Learn building an optimization model, feasible inputs.

Lesson 1 of 32 parts Practice Flashcards

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Building an Optimization Model

Objective: Explain building an optimization model, apply the stated method, and check its conditions.

Optimization begins by expressing the objective in one variable using the constraints. For rectangle perimeter P, the other side is P/2−x, so area is x(P/2−x). Both dimensions must be positive before an interior candidate is considered. State the feasible interval before optimizing.

Formulas

$A(x)=x(P/2−x)$

Worked examples

Try itA rectangle has perimeter 28 cm. Write area as A(x)=x(14−x). Find A(6) in cm².Show answer

Step 1: The other side is half the perimeter minus x. Step 2: At x=6, the other side is 8>0. Step 3: Area is 48 cm².

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Building an Optimization Model

Objective: Explain building an optimization model, apply the stated method, and check its conditions.

Formulas

$A(x)=x(P/2−x)$

Worked examples

Try itA rectangle has perimeter 28 cm. Write area as A(x)=x(14−x). Find A(6) in cm².Show answer

Step 1: The other side is half the perimeter minus x. Step 2: At x=6, the other side is 8>0. Step 3: Area is 48 cm².

Feasible Inputs

Objective: Explain feasible inputs, apply the stated method, and check its conditions.

The feasible domain contains inputs satisfying every physical and mathematical constraint. Positive rectangle dimensions require 0<x<P/2 for fixed perimeter P. A stationary point outside that interval is not a possible design, regardless of its algebraic behavior. An algebraic candidate outside the feasible domain must be excluded.

Formulas

0<x<P/2 for positive fixed-perimeter rectangle sides

Worked examples

Try itA rectangle has perimeter 28 cm. For A=x(14−x), what is the excluded upper endpoint of the feasible x interval?Show answer

Step 1: Both side lengths must be positive. Step 2: Require x>0 and 14−x>0. Step 3: The feasible interval is (0,14); the upper endpoint is 14.

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