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Ch. 4 - Applications of the Derivative
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 4 - Applications of the DerivativeProblem 4.5.77
Chapter 4, Problem 4.5.77

Tree notch (Putnam Exam 1938, rephrased) A notch is cut in a cylindrical vertical tree trunk (see figure). The notch penetrates to the axis of the cylinder and is bounded by two half-planes that intersect on a diameter D of the tree. The angle between the two half-planes is Θ. Prove that for a given tree and fixed angle Θ, the volume of the notch is minimized by taking the bounding planes at equal angles to the horizontal plane that also passes through D.

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Understand the geometry of the problem: The notch is formed by two half-planes intersecting along a diameter of the cylindrical tree trunk. The angle between these planes is Θ, and they penetrate to the axis of the cylinder.
Visualize the problem: Imagine the cylinder with its axis vertical. The diameter D is a horizontal line through the center of the cylinder. The two half-planes intersect along this diameter, forming a wedge-like notch.
Consider the orientation of the half-planes: The problem states that the volume of the notch is minimized when the bounding planes are at equal angles to the horizontal plane passing through D. This suggests symmetry in the configuration of the planes.
Set up the mathematical model: Use cylindrical coordinates (r, θ, z) to describe the geometry. The notch volume can be expressed as an integral over the region defined by the intersection of the half-planes and the cylinder. The symmetry implies that the angles of the planes with respect to the horizontal should be equal, i.e., each plane makes an angle of Θ/2 with the horizontal.
Calculate the volume: The volume of the notch can be found by integrating over the region defined by the intersection of the half-planes and the cylinder. The symmetry condition simplifies the integration, as the region is symmetric about the diameter D. The integral will involve the radius of the cylinder and the angle Θ, and the symmetry condition ensures that the volume is minimized.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Volume of a Solid of Revolution

The volume of a solid of revolution is calculated using integral calculus, specifically the disk or washer method. This involves rotating a region around an axis to create a three-dimensional shape. Understanding how to set up these integrals is crucial for determining the volume of the notch in the tree, as it involves geometric considerations of the cylindrical shape.
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Optimization in Calculus

Optimization involves finding the maximum or minimum values of a function. In this context, we need to minimize the volume of the notch by adjusting the angles of the bounding planes. Techniques such as taking derivatives and applying the first and second derivative tests are essential for identifying optimal angles that yield the minimum volume.
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Geometric Interpretation of Angles

The geometric interpretation of angles is vital for understanding how the bounding planes interact with the tree trunk. The angle Θ between the two half-planes affects the shape and volume of the notch. Recognizing how these angles relate to the horizontal plane and the diameter D helps in visualizing the problem and applying calculus effectively to find the solution.
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