MathLab with Mr. Ahmed

MathLab with Mr. Ahmed is a math learning website by Ahmed Amasha, a mathematics teacher teaching the American curriculum. Explore Algebra 2, Precalculus, Calculus, and SAT and PSAT Math Prep through lessons, focused practice and printable resources.

MathLabwith Mr. Ahmed
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2.M Focus on Modeling (Chapter Supplement)

Build, test, and optimize function models from geometric and real-world constraints.

Before you solve

Ideas in this lesson

Explore the lesson brief and the worked example for Idea 11. Create a free account to continue with the remaining ideas and practice.

01Translate geometric dimensions into one-variable modelsWorked steps · five questions02Recover dimensions and convert geometric measuresWorked steps · five questions03Build and interpret similar-triangle shadow modelsWorked steps · five questions04Model separation in perpendicular motionWorked steps · five questions05Model constrained triangle and semicircle areasWorked steps · five questions06Optimize products and sums of squares under a fixed sumWorked steps · five questions07Optimize area with available fencingWorked steps · five questions08Minimize fencing and cost with fixed areaWorked steps · five questions09Optimize areas made from split wireWorked steps · five questions10Optimize rectangle-and-semicircle windowsWorked steps · five questions11Model and optimize corner-cut open boxesWorked steps · five questions12Minimize material for fixed-volume containersWorked steps · five questions13Optimize rectangles inscribed under parabolasWorked steps · five questions14Minimize mixed-speed travel time and energyWorked steps · five questions15Derive and optimize a kite from its border lengthsWorked steps · five questions
Lesson brief and rules

2.M Focus on Modeling (Chapter Supplement)

Identify the quantity to model, choose a variable, express the constraints, state a realistic domain, and interpret the model with units. Compare graphical or numerical evidence with justified conclusions.

Remember
An algebraic result must satisfy the original geometric and contextual constraints.
Worked example 11
01 / A rectangular habitat is 22 times as long as it is wide. Let xx measure its width in meters. Derive a one-variable model, and state the physically meaningful domain and output units. Find the width needed for an area of 1818 square meters; explain why a negative algebraic root is rejected. Complete the written work or drawing first. The selected answer checks the result only; your reasoning or drawing needs teacher/self-review.
02 / Length is 22 times width, so multiply xx by 2x2x.
03 / Solve 2x22 x^{2}=18=18. The roots have opposite signs; only 33 meters is a physical width.
Worked example 2 · Translate geometric dimensions into one-variable models

A display board is 44 centimeters longer than its width. Let xx measure its width in centimeters. Derive a one-variable model, and state the physically meaningful domain and output units. Evaluate the area at widths 4,64, 6, and 88 centimeters and explain why area is not proportional to width. Complete the written work or drawing first. The selected answer checks the result only; your reasoning or drawing needs teacher/self-review.

Length equals x+4x+4, and area is width times length.
Substitute 4,64, 6, and 88 into x(x+4)x \left(x + 4\right). The ratio of area to width changes with width.
Written-work review:

15 questions with a worked answer for each.Start practice ↗
Textbook reference: Stewart, Redlin & Watson, Precalculus: Mathematics for Calculus, 7th ed. (Cengage, 2016). The explanations and practice here are original, organized by textbook section.

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