Find the length of the entire perimeter of the region inside

Question

r = 16sin(theta) but outside r = 4.

Answer 1

First define the region. Drawing a sketch chould help.

16 sin*theta = 4 when
theta = arcsin 1/4
which cporresponds to 14.48 degrees and 165.52 degrees

Your region is the area bounded by 16 sin(theta) on the outside and r = 4 on the inside. The outer curve is called a cardioid, as I recall. There is a fomula for the arc length of a cardioid at
http://en.wikipedia.org/wiki/Cardioid

The length of the circular arc is trivial. You will need to keep theta in radians.

Answer 2

20*(2.888912398 - .25268002551)

Answer 3

64π