Calculus I · 2B · lesson
Geometric Optimization
Learn geometric optimization with clear exposition, guided derivations, worked examples, visuals, checks, and interpretation.
Section overview
OptimizationWhat this section is building
Learn geometric optimization with clear exposition, guided derivations, worked examples, visuals, checks, and interpretation.
Optimization is a modeling problem first: the derivative only compares candidates after the geometry, units, and feasible domain are correct.
Write variables and units, identify the objective, use the constraint to eliminate a variable, then test critical and boundary candidates.
Optimizing the constraint, ignoring the feasible domain, or keeping an algebraic critical point that cannot exist in the real design.
Learning objectives
Solve geometric optimization problems with Pythagorean and area constraints.
Distance, Area, and Volume Problems
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.
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.
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.
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.
Closest point on a parabola
Find the point on closest to .
Worked solution
Write a real attempt before opening the supplied answer.
Largest rectangle under a curve
If the upper corners of a rectangle lie on and the rectangle is symmetric about the -axis, width is , height is , and area is
The geometry determines both the objective and the domain.
Run cable across land and underwater
A station lies km offshore from the nearest point on a straight coast. A facility lies km down the coast from . Underwater cable costs three times as much per kilometer as land cable. If the cable comes ashore km from , the cost model is proportional to
Differentiate, solve , and compare with the endpoints. This is a genuine tradeoff: a longer land route can reduce expensive underwater distance.
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