Consider a plane curve defined by the parametric equations : x(t)=acos^3 (t) and y(t)=asin^3 (t) where a is a constant and 0≤t≤2π. This curve is known as the astroid. Find the curvature κ(t) of this curve at any point t. Specifically, calculate the curvature at t= π/4 .

Consider a plane curve defined by the parametric equations : x(t)=acos^3 (t) and y(t)=asin^3 (t) where a is a constant and 0≤t≤2π. This curve is known as the astroid. Find the curvature κ(t) of this curve at any point t. Specifically, calculate the curvature at t= π/4 .

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Consider a plane curve defined by the parametric equations :
x(t)=acos^3 (t) and y(t)=asin^3 (t)
where a is a constant and 0≤t≤2π.
This curve is known as the astroid.

Find the curvature κ(t) of this curve at any point t.
Specifically, calculate the curvature at t= π/4 .

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Step 1: Conceptual Introduction

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Step 2: Calculating the First and Second Derivatives

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Step 3: Substituting into the Curvature Formula

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Step 4: Simplifying and Calculating the Curvature at π/4

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