Chapter 8.3, Problem 109E

(a)

To determine

To write: a formula in series notation that gives the surface area of the sphere flake.

Answer to Problem 109E

The formula in series notation that gives the surface area of the sphere flake we have to find a geometric series that models the situation is given by

  ${\displaystyle \sum _{n=1}^{\infty }4\pi }$

Explanation of Solution

Given information:

The radius of the large sphere is 1. Attached to the large sphere are nine spheres of radius $\frac{1}{3}$ . Attached to the large sphere are nine spheres of radius $\frac{1}{3}$ . Attached to given of the smaller spheres are nine spheres of radius $\frac{1}{9}$ .

  

Calculation:

To determine a formula in series notation that gives the surface area of the sphere flake we have to find a geometric series that models the situation as shown below:

  $\begin{array}{c}{A}_1=4\pi r_1{}^2\\ =4\pi {\left(1\right)}^2\\ =4\pi \\ A_2=4\pi r_2{}^2\\ =4\pi {\left(\frac{1}{3}\right)}^2\\ =9\left(4\pi {\left(\frac{1}{3}\right)}^2\right)\\ =4\pi \end{array}$

Continue to determine the surface areas as above we get

  $\begin{array}{c}{A}_3=4\pi r_3{}^2\\ =4\pi {\left(\frac{1}{9}\right)}^2\\ =81\left(4\pi {\left(\frac{1}{9}\right)}^2\right)\\ =4\pi \end{array}$

Now, the formula in series notation that gives the surface area of the sphere flake we have to find a geometric series that models the situation is given by

  $\begin{array}{c}S=A_1+A_2+A_3+\mathrm{...}\\ =4\pi +4\pi +4\pi +\mathrm{...}\\ ={\displaystyle \sum _{n=1}^{\infty }4\pi }\end{array}$

(b)

To determine

To write: a formula in series notation that gives the volume of the sphere flake.

Answer to Problem 109E

The formula in series notation that gives the volume of the sphere flake we have to find a geometric series that models the situation is given by

  ${\displaystyle \sum _{n=1}^{\infty }\left(\frac{4\pi }{3^n}\right)}$

Explanation of Solution

Given information:

The radius of the large sphere is 1. Attached to the large sphere are nine spheres of radius $\frac{1}{3}$ . Attached to the large sphere are nine spheres of radius $\frac{1}{3}$ . Attached to given of the smaller spheres are nine spheres of radius $\frac{1}{9}$ .

  

Calculation:

To determine a formula in series notation that gives the volume of the sphere flake we have to find a geometric series that models the situation as shown below:

  $\begin{array}{c}{V}_1=\frac{4}{3}\pi r_1{}^3\\ =\frac{4}{3}\pi {\left(1\right)}^3\\ =\frac{4}{3}\pi \end{array}$

Continue to determine the surface areas as above we get

  $\begin{array}{c}{V}_2=\frac{4}{3}\pi r_2{}^3\\ =\frac{4}{3}\pi {\left(\frac{1}{3}\right)}^3\\ =9\left(\frac{4}{3}\pi {\left(\frac{1}{3}\right)}^3\right)\\ =\frac{4}{9}\pi \end{array}$

Continue to determine the surface areas as above we get

  $\begin{array}{c}{V}_3=\frac{4}{3}\pi r_3{}^3\\ =\frac{4}{3}\pi {\left(\frac{1}{9}\right)}^3\\ =81\left(\frac{4}{3}\pi {\left(\frac{1}{}\right)}^3\right)\\ =\frac{4\pi }{27}\end{array}$

Now, the formula in series notation that gives the volume of the sphere flake we have to find a geometric series that models the situation is given by

  $\begin{array}{c}V=V_1+V_2+V_3+\mathrm{...}\\ =\frac{4\pi }{3}+\frac{4\pi }{9}+\frac{4\pi }{27}+\mathrm{...}\\ ={\displaystyle \sum _{n=1}^{\infty }\left(\frac{4\pi }{3^n}\right)}\end{array}$

(c)

To determine

Whether the surface area and the volume of the sphere flake finite or infinite. To find the value of finite surface area and volume.

Answer to Problem 109E

The volume is $2\pi \left(1-{\left(\frac{1}{3}\right)}^n\right)$ .

Explanation of Solution

Given information:

The radius of the large sphere is 1. Attached to the large sphere are nine spheres of radius $\frac{1}{3}$ . Attached to the large sphere are nine spheres of radius $\frac{1}{3}$ . Attached to given of the smaller spheres are nine spheres of radius $\frac{1}{9}$ .

  

Calculation:

The series formula for finding the surface area of the sphere flake is $S={\displaystyle \sum _{n=1}^{\infty }4\pi }$ and the series formula for finding the volume is $V={\displaystyle \sum _{n=1}^{\infty }\left(\frac{4\pi }{3^n}\right)}$ .

From these two formulae we conclude that the surface area of the sphere flake is infinite and the volume is finite if the process is continued infinitely.

That means,

  $\begin{array}{c}S={\displaystyle \sum _{n=1}^{\infty }4\pi }\\ \to \infty \text{square}\text{units}\\ V={\displaystyle \sum _{n=1}^{\infty }\left(\frac{4\pi }{3^n}\right)}\\ =2\pi \text{cubic}\text{units}\end{array}$

On the other hand the surface area of the sphere flake is finite and the volume is also finite if the process is continued finitely.

If the process of attaching spheres is continued upto $n$ times then the surface area and the volume of the sphere flake are given by

  $\begin{array}{c}S={\displaystyle \sum _{i=1}^{n}4\pi }\\ =4\pi {\displaystyle \sum _{i=1}^n\left(1\right)}\\ =4\pi \left(1+2+3+\mathrm{...}+n\right)\\ =2\pi n\left(n+1\right)\text{square}\text{units}\end{array}$

The volume is given by

  $\begin{array}{c}V={\displaystyle \sum _{i=1}^n\frac{4\pi }{3}{\left(\frac{1}{3}\right)}^{n-1}}\\ =\frac{4\pi }{3}\left(\frac{1-{\left(\frac{1}{3}\right)}^n}{1-\left(\frac{1}{3}\right)}\right)\\ =\frac{4\pi }{3}\left(1-{\left(\frac{1}{3}\right)}^n\right)\left(\frac{3}{2}\right)\\ =2\pi \left(1-{\left(\frac{1}{3}\right)}^n\right)\end{array}$