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## 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.

**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* = *aθ*

2 = *aπ*

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.

**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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