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.

optimization problems

Optimization Problems

Optimization problems often involve identifying maximum or minimum values under specific constraints. A well-known example is determining the longest horizontal pipe that can be moved around a right-angled corner, where a 3-meter-wide hallway meets a 2-meter-wide hallway. This scenario, common in architectural design and industrial transport, can be understood conceptually through geometric and trigonometric reasoning.To visualize the problem, consider the pipe as a straight line that touches the inner corner of the turn and extends outward to touch the opposite walls of each hallway. The total length of the pipe depends on its orientation, defined by the angle it forms with the walls. For any given angle, the pipe must simultaneously clear both hallways, and its length is constrained by the tightest section of the corner it passes through.Rather than attempting to directly find the maximum possible length, the problem is reframed by considering the shortest possible clearance path the pipe can take. This minimum clearance corresponds to the most restrictive position in which the pipe can still navigate the corner. Calculus is then applied to identify this critical point by analyzing how the total path length changes with angle. Although the detailed steps involve differentiation and trigonometric identities, the central idea is to locate the angle that yields the least clearance, which in turn determines the maximum allowable pipe length. To find the pipe length that works at all angles, we minimize L(&#952;). This ensures we identify the minimum of the longest possible lengths&#8212;i.e., the greatest pipe length that fits, no matter the approach angle.This approach illustrates how minimizing a function&#8212;rather than directly maximizing the quantity of interest&#8212;can provide a solution in constrained optimization settings. The final result gives a precise value for the longest pipe that can successfully round the corner without vertical tilting.

Optimization problems often involve identifying maximum or minimum values under specific constraints. A well-known example is determining the longest ...

Education

JoVE Core

Calculus

Unit

Applications of Differentiation

KEY TERMS AND CONCEPTS

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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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...

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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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