Question

A sphereflake is a fractal constructed in an infinite process. In a certain sphereflake, a large sphere has a radius of 1. Attached to this are 16 smaller spheres, each with radius 1/4 . Attached to each of these 16 smaller spheres are 16 even smaller spheres with a radius of 1/16 and so on. The surface area of a sphere is given by the formula A=4πr^2 , and the volume of a sphere is given by the formula V=4/3πr^3 . Determine infinite geometric series for the total surface area and the total volume of the sphereflake. Determine whether both, only one, or neither of the series converges.

Answer 1

is this true?

Answer 2

A sphereflake is a fractal constructed in an infinite process. In a certain sphereflake, a large sphere has a radius of 1. Attached to this are 4 smaller spheres, each with radius 1/2 . Attached to each of these 4 smaller spheres are 4 even smaller spheres with a radius of 1/4 and so on. Using the fact that the volume of a sphere is given by the formula V=4/3πr^3 , find the total volume of the sphereflake if it is defined.

Answer 3

Answers for da math
1) 160
2) 3,214
3)204
4)only the geometric series for the total surface area converges
5) 8/3 pi