Name: pipe length around a corner
Uploaded: 2026-01-12T12:24:22.000Z
Duration: 1 min 26 s
Channel: JoVE

A practical example of optimization involves determining the maximum length of a rod that can be carried around a right-angle corner formed by a 3-meter-wide hallway and a 2-meter-wide hallway, without tilting it vertically.To solve this, imagine a line segment passing through the inner corner and touching the outer walls. This segment represents the available clearance at a specific angle.This length L is divided into two components, L1&#8203; and L2, which can be written in terms of the hallway widths and the sine and cosine of the angle.While the goal is to find the maximum length, this length is limited by the tightest part of the turn.So, differentiate the length function to find where the slope is zero, identifying the minimum clearance that acts as a bottleneck for the rod.The resulting equation can be solved by rewriting the secant and cosecant terms as sines and cosines. Next, rearranging the terms to opposite sides of the equation to group the sines and cosines gives a simplified expression involving the tangent cubed.Substituting this angle back into the original length equation provides the maximum length of the rod that can safely clear the corner.

optimalisatieproblemen

Optimization Problems

Optimalisatieproblemen hebben vaak betrekking op het bepalen van maximale of minimale waarden onder specifieke beperkingen. Een bekend voorbeeld is het vaststellen van de langste horizontale pijp die rond een rechte hoek kan worden verplaatst, waarbij een gang van 3 m breed aansluit op een gang van 2 m breed. Dit scenario, dat vaak voorkomt in de architectuur en het industri&#xEB;le transport, kan conceptueel worden begrepen aan de hand van meetkundige en trigonometrische redeneerstappen.
Om het probleem te visualiseren, kan men de pijp beschouwen als een rechte lijn die de binnenhoek van de bocht raakt en zich uitstrekt tot aan de tegenoverliggende wanden van beide gangen. De totale lengte van de pijp hangt af van de ori&#xEB;ntatie, die wordt bepaald door de hoek die de pijp met de wanden vormt. Voor elke gegeven hoek moet de pijp gelijktijdig door beide gangen kunnen passen, waarbij de lengte wordt begrensd door de nauwste passage van de bocht waar de pijp doorheen moet.
In plaats van rechtstreeks te proberen de maximaal mogelijke lengte te bepalen, wordt het probleem herformuleerd door te kijken naar de kleinst mogelijke speling die de pijp kan volgen. Deze minimale speling correspondeert met de meest beperkende positie waarin de pijp de hoek nog net kan passeren. Vervolgens wordt differentiaalrekening toegepast om dit kritieke punt te identificeren door te analyseren hoe de totale lengte verandert met de hoek. Hoewel de gedetailleerde uitwerking gebruikmaakt van differenti&#xEB;ren en trigonometrische identiteiten, is het centrale idee het bepalen van de hoek die de kleinste speling oplevert, hetgeen op zijn beurt de maximaal toelaatbare pijplengte vastlegt. Om de pijplengte te vinden die voor alle hoeken voldoet, minimaliseert men L(&#x3B8;). Daarmee wordt de minimale waarde van de maximaal mogelijke lengtes bepaald, oftewel de grootste pijplengte die ongeacht de benaderingshoek past.
Deze benadering illustreert hoe het minimaliseren van een functie, in plaats van het direct maximaliseren van de grootheid van belang, een effectieve oplossing kan bieden in optimalisatieproblemen met beperkingen. Het eindresultaat levert een exacte waarde op voor de langste pijp die de bocht succesvol kan nemen zonder verticaal te kantelen.

Optimalisatieproblemen

Optimalisatieproblemen hebben vaak betrekking op het bepalen van maximale of minimale waarden onder specifieke beperkingen. Een bekend voorbeeld is het ...

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Applications of Differentiation

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KEY TERMS AND CONCEPTS

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pipe length around a corner

Pipe length around a corner is a classic constrained optimization problem. It shows how calculus can find a maximum or minimum value when space is limited by fixed walls and angles.

The example used here asks for the longest horizontal pipe that can move around a right-angled turn. One hallway is 3 meters wide and the other is 2 meters wide. The pipe is shown as a straight line that touches the inner corner and reaches the opposite walls of both hallways.

The key idea is that the pipe’s length changes with its angle. For each angle, the pipe must still clear both hallways. That means the tightest part of the corner limits how long the pipe can be.

Instead of trying to maximize the pipe length directly, the problem is rewritten as a search for the shortest clearance path. The minimum clearance gives the most restrictive position where the pipe can still pass. Calculus then helps find this critical angle by looking at how the length function L(θ) changes with θ.

The solution uses differentiation and trigonometric identities, but the main idea is simple. Find the angle that gives the least clearance, and that angle sets the maximum pipe length. This approach shows how minimizing a function can solve a constrained optimization problem.

Pipe Length Around a Corner

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key terms and concepts

Learn how calculus and trigonometry solve a pipe-length optimization problem by finding the longest horizontal pipe that can turn a 3 m by 2 m hallway corner.

Pipe Length Around a Corner | JoVE K12

The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

calculus

Pipe Length Around a Corner

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