Part A: The polar curve represented by r1 = 3 + 3sin θ is a limaçon. To see this, we can simplify the equation as follows:
r1 = 3 + 3sin θ
r1 - 3 = 3sin θ
(r1 - 3)/3 = sin θ
This is the equation of a limaçon with an inner loop when (r1 - 3)/3 < 1, and an outer loop when (r1 - 3)/3 > 1. In this case, (r1 - 3)/3 = sin θ, which means that the curve has an inner loop since sin θ ranges between -1 and 1. Therefore, the polar curve represented by r1 = 3 + 3sin θ is a limaçon with an inner loop.
Part B: The curve is symmetrical to the polar axis. To see this, we can substitute -θ for θ in the equation r1 = 3 + 3sin θ, which gives:
r1 = 3 + 3sin(-θ)
r1 = 3 - 3sin θ
Since the equation of the curve is unchanged by replacing θ with -θ, the curve is symmetrical to the polar axis.
Part C: The two main differences between the graphs of r1 = 3 + 3sin θ and r2= 8 + 3cos θ are:
1. Shape: The two curves have different shapes. The curve represented by r1 = 3 + 3sin θ is a limaçon with an inner loop, while the curve represented by r2 = 8 + 3cos θ is a circle with radius 3 centered at (0, 8) in the polar plane.
2. Orientation: The two curves are oriented differently in the polar plane. The curve represented by r1 = 3 + 3sin θ has its inner loop oriented towards the origin, while the curve represented by r2 = 8 + 3cos θ is centered at (0, 8) and is oriented vertically along the polar axis.
These differences are due to the different trigonometric functions used in the equations of the curves. The curve represented by r1 = 3 + 3sin θ is based on the sine function, while the curve represented by r2 = 8 + 3cos θ is based on the cosine function.