   Chapter 10, Problem 13PS

Chapter
Section
Textbook Problem

Finding a Polar Equation Four dogs are located at the corners of a square with sides of length d. The dogs all move counterclockwise at the same speed directly toward the next dog, as shown in the figure. Find the polar equation of a dog’'s path as it spirals toward the center of the square. To determine

To calculate: The polar equation that describes the dog’s path.

Explanation

Given: Four dogs located at the corners of the square of length d move towards their neighbor in the anti-clockwise direction as shown:

Formula used:

The slope of the tangent line of the polar curve r=f(θ) at the point (r,θ) is:

m=f(θ)cosθf'(θ)sinθf(θ)sinθ+f'(θ)cosθ

Calculation:

Consider the dog located in the first quadrant.

The coordinates of the dog are (r,θ).

The coordinates of its neighbor to the left are (r,θ+π2).

Convert the polar coordinates into the Cartesian coordinates:

(x1,y1)=(rcosθ,rsinθ)(x2,y2)=(rcos(θ+π2),rsin(θ+π2))=(rsinθ+rcosθ)

The slope of line joining the two points willbe:

m=rcosθrsinθrsinθrcosθ=sinθcosθsinθ+cosθ

This should be equal to the slope of the tangent line at the point (r,θ)

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