Math Problem Statement

Find the radius of curvature of the following curves: a) π‘₯ 3 + 𝑦 3 = 3π‘Žπ‘₯𝑦 at the point ( 3π‘Ž 2 , 3π‘Ž 2 ), where π‘Ž is constant. b) π‘Ÿ = π‘Ž (1 βˆ’ cos πœƒ) at πœƒ = πœ‹ 2

Solution

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Math Problem Analysis

Mathematical Concepts

Differential Calculus
Curvature
Polar Coordinates
Implicit Differentiation

Formulas

Radius of Curvature in Cartesian Coordinates: R = (1 + (dy/dx)^2)^(3/2) / |d^2y/dx^2|
Radius of Curvature in Polar Coordinates: R = (r^2 + (dr/dΞΈ)^2)^(3/2) / |r^2 + 2(dr/dΞΈ)^2 - r(d^2r/dΞΈ^2)|

Theorems

Differentiation Rules for Implicit Functions
Radius of Curvature Theorem

Suitable Grade Level

Undergraduate Calculus (College Level)