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2. Show that an equilateral triangle can be inscribed in the epitrochoid and that its centroid is on the circle of radius b centered at the origin.

2. Show that an equilateral triangle can be inscribed in the epitrochoid and that its centroid is on the circle of radius b centered at the origin.

Algebra: Structure And Method, Book 1
Algebra: Structure And Method, Book 1
(REV)00th Edition
ISBN:9780395977224
Author:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole
Publisher:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole
Chapter4: Polynomials
Section4.9: Area Problems
Problem 10P
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Just #2. Thank you. 

Problems 2 – 3. A circle C of radius 2r has its center at the origin. A circle of radius r rolls without slipping in the counterclockwise direction around C. A point P is located on a fixed radius of the rolling circle at a distance b from its center, 0 < b <r. Let L be the line from the center of C to the center of the rolling circle and t be the angle that L makes with the positive x-axis. Using t as a parameter we will see in a video (coming soon) that parametric equations of the path traced out by P are x(t) = 3r cos t + b cos 3t and y(t) = 3r sin t + b sin 3t Note: The path traced out by P is called an epitrochoid. 2. Show that an equilateral triangle can be inscribed in the epitrochoid and that its centroid is on the circle of radius b centered at the origin. Note: This is the principle of the Wankel rotary engine. When the equilateral triangle rotates with its vertices on the epitrochoid, its centroid sweeps out a circle whose center is at the center of the curve. 3. In most rotary engines the sides of the equilateral triangles are replaced by arcs of circles centered at the opposite vertices, thus the diameter of the rotor is constant. Show that the rotor will fit in the epitrochoid if b <(2 – V3)r.
Transcribed Image Text:Problems 2 – 3. A circle C of radius 2r has its center at the origin. A circle of radius r rolls without slipping in the counterclockwise direction around C. A point P is located on a fixed radius of the rolling circle at a distance b from its center, 0 < b <r. Let L be the line from the center of C to the center of the rolling circle and t be the angle that L makes with the positive x-axis. Using t as a parameter we will see in a video (coming soon) that parametric equations of the path traced out by P are x(t) = 3r cos t + b cos 3t and y(t) = 3r sin t + b sin 3t Note: The path traced out by P is called an epitrochoid. 2. Show that an equilateral triangle can be inscribed in the epitrochoid and that its centroid is on the circle of radius b centered at the origin. Note: This is the principle of the Wankel rotary engine. When the equilateral triangle rotates with its vertices on the epitrochoid, its centroid sweeps out a circle whose center is at the center of the curve. 3. In most rotary engines the sides of the equilateral triangles are replaced by arcs of circles centered at the opposite vertices, thus the diameter of the rotor is constant. Show that the rotor will fit in the epitrochoid if b <(2 – V3)r.
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