Concept explainers
(a)
To write: a formula in series notation that gives the surface area of the sphere flake.
(a)
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
Explanation of Solution
Given information:
The radius of the large sphere is 1. Attached to the large sphere are nine spheres of radius
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:
Continue to determine the surface areas as above we get
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
(b)
To write: a formula in series notation that gives the volume of the sphere flake.
(b)
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
Explanation of Solution
Given information:
The radius of the large sphere is 1. Attached to the large sphere are nine spheres of radius
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:
Continue to determine the surface areas as above we get
Continue to determine the surface areas as above we get
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
(c)
Whether the surface area and the volume of the sphere flake finite or infinite. To find the value of finite surface area and volume.
(c)
Answer to Problem 109E
The volume is
Explanation of Solution
Given information:
The radius of the large sphere is 1. Attached to the large sphere are nine spheres of radius
Calculation:
The series formula for finding the surface area of the sphere flake is
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,
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
The volume is given by
Chapter 8 Solutions
Precalculus with Limits: A Graphing Approach
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