Chapter 10, Problem 13PS

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347

Chapter
Section

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347
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(Î¸) at the point (r,Î¸) is:

m=f(Î¸)cosÎ¸âˆ’f'(Î¸)sinÎ¸âˆ’f(Î¸)sinÎ¸+f'(Î¸)cosÎ¸

Calculation:

Consider the dog located in the first quadrant.

The coordinates of the dog are (r,Î¸).

The coordinates of its neighbor to the left are (r,Î¸+Ï€2).

Convert the polar coordinates into the Cartesian coordinates:

(x1,y1)=(rcosÎ¸,rsinÎ¸)(x2,y2)=(rcos(Î¸+Ï€2),rsin(Î¸+Ï€2))=(âˆ’rsinÎ¸+rcosÎ¸)

The slope of line joining the two points willbe:

m=rcosÎ¸âˆ’rsinÎ¸âˆ’rsinÎ¸âˆ’rcosÎ¸=sinÎ¸âˆ’cosÎ¸sinÎ¸+cosÎ¸

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

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