Math Problem Statement
If you want to roll a ball down at rack in the shortest time possible, what shape should the track make? A straight line? A semi-circle? This is a famous physics questions with a surprising answer. The “curve of fastest decent” is called a brachistochrone, and is the same shape that a single point on a rolling wheel traces. It has this parametric equation: x(t) = t − sin(t), y(t) = cos(t) − 1. At what points does the brachistochrone have a horizontal tangent?
Solution
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Math Problem Analysis
Mathematical Concepts
Parametric Equations
Derivatives
Cycloid
Tangent Line
Formulas
x(t) = t - sin(t)
y(t) = cos(t) - 1
dy/dx = (dy/dt) / (dx/dt)
Theorems
Chain Rule of Differentiation
Brachistochrone Problem
Suitable Grade Level
University Level (Calculus)
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