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Geometry and Polar Coordinates Practice Problems

— McGraw-Hill Professional
Updated on Oct 3, 2011

If necessary, review:

Geometry and Polar Coordinates Practice Problems

PROBLEM 1

What is the value of the constant, a , in the spiral shown in Fig. 10-10? What is the equation of this spiral? Assume that each radial division represents 1 unit.

 

 

Alternative Coordinates Some Examples Cardioid

Fig. 10-10 . Polar graph of a spiral; illustration for Problem 1

SOLUTION 1

Note that if θ = π, then r = 2. Therefore, we can solve for a by substituting this number pair in the general equation for the spiral. We know that ( θ , r ) = (π,2), and that is all we need. Proceed like this:

r =

2 =

2/π = a

Therefore, a = 2/π , and the equation of the spiral is r = (2/π) θ or, in a somewhat simpler form without parentheses, r = 2 θ /π.

PROBLEM 2

What is the value of the constant, a , in the cardioid shown in Fig. 10-11? What is the equation of this cardioid? Assume that each radial division represents 1 unit.

Alternative Coordinates Some Examples Cardioid

Fig. 10-11 . Polar graph of a cardioid; illustration for Problem 2

SOLUTION 2

Note that if θ = 0, then r = 4. We can solve for a by substituting this number pair in the general equation for the cardioid. We know that ( θ , r ) = (0,4), and that is all we need. Proceed like this:

r = 2 a (1 + cos θ )

4 = 2 a (1 + cos 0)

4 = 2 a (1 + 1)

4 = 4 a

a = 1

This means that the equation of the cardioid is r = 2(1 + cos θ ) or, in a simpler form without parentheses, r = 2 + 2 cos θ .

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