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Calculus I · Unit 2B · lesson

Geometric Optimization

Concept

Learning objectives

Solve geometric optimization problems with Pythagorean and area constraints.

Distance, Area, and Volume Problems

Explanation

Before the formulas

Optimization in Geometric Optimization rewards a labeled diagram and punishes vague variables. Define each dimension in words, derive rather than guess the objective formula, and state how the constraint links the dimensions. If the model includes cost, revenue, dosage, or material, explain what assumptions make that formula plausible.

After finding an optimum, perform a reality check. Negative lengths, impossible production levels, or parameter values outside the model's range are not rescued by correct calculus. The best mathematical candidate must also be feasible in the stated situation.

Cutting squares of side x from each corner leaves base dimensions L-2x and W-2x and creates height x. The volume model comes directly from the folded geometry.
Read this graph as text

A box-from-a-sheet problem begins with the net. Cutting squares of side x from each corner leaves base dimensions L-2x and W-2x and creates height x . The volume model comes directly from the folded geometry. The cut size x becomes the box height. Each original sheet dimension loses x at both ends, which explains the factors L-2x and W-2x . The feasible domain is limited by the smaller sheet dimension; otherwise a base length becomes zero or negative.

Every relationship in a box-from-a-sheet problem begins with the net is identified with written labels plus distinct solid, dashed, dotted, double, marker, or pattern cues; color is never the only carrier of meaning.

Why it matters: The net is essential because the volume formula is otherwise easy to memorize incorrectly. It also makes the physical domain visible.

Visual study

Cutting squares of side x from each corner leaves base dimensions L-2x and W-2x and creates height x. The volume model comes directly from the folded geometry.

Explanation

A good diagram prevents a bad objective function

Label every changing dimension and mark which quantities are fixed. For boxes, fences, cylinders, and distance problems, the picture determines the constraint and helps expose impossible values.

After finding a candidate, return to the diagram. The numbers should fit the geometry and should have sensible units. This final check catches many algebraically polished but physically impossible answers.

Geometric optimization converts spatial constraints into algebra. Similar triangles, the Pythagorean theorem, perimeter formulas, and volume relations reduce the design to one variable. A well-labeled diagram is often worth more than a page of hurried differentiation.

The final answer must include dimensions and units, not merely the critical input. Verify that the proposed design satisfies the original constraint and makes physical sense.

Guided walkthrough

Closest point on a parabola

Find the point on y=x2y=x^2 closest to (0,3)(0,3).

Answer reveal

Worked solution

Write a real attempt before opening the supplied answer.

Worked example

Largest rectangle under a curve

If the upper corners of a rectangle lie on y=12−x2y=12-x^2 and the rectangle is symmetric about the yy-axis, width is 2x2x, height is 12−x212-x^2, and area is

A(x)=2x(12−x2),0≤x≤12.A(x)=2x(12-x^2), \qquad0\le x\le\sqrt{12}.

The geometry determines both the objective and the domain.

Modeling lab

Run cable across land and underwater

A station lies 88 km offshore from the nearest point AA on a straight coast. A facility lies 2020 km down the coast from AA. Underwater cable costs three times as much per kilometer as land cable. If the cable comes ashore xx km from AA, the cost model is proportional to

C(x)=3x2+64+(20−x),0≤x≤20.C(x)=3\sqrt{x^2+64}+(20-x), \qquad0\le x\le20.

Differentiate, solve C′(x)=0C'(x)=0, and compare with the endpoints. This is a genuine tradeoff: a longer land route can reduce expensive underwater distance.

After the explanation

Use the section idea

Reading lens

Separate the objective from the constraint, reduce to one feasible variable, and interpret the winning candidate in the original design.

Mental model

Optimization is a modeling problem first: the derivative only compares candidates after the geometry, units, and feasible domain are correct.

Decision

Write variables and units, identify the objective, use the constraint to eliminate a variable, then test critical and boundary candidates.

Common trap

Optimizing the constraint, ignoring the feasible domain, or keeping an algebraic critical point that cannot exist in the real design.

Check yourself

Have you compared every feasible candidate and explained why the result is physically and economically reasonable?

Source & rights

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