Concept explainers
Geometry A circular disk of radius R is cut out of paper, as shown in figure (a). Two disks of radius
To find: The total area of all disks which radiuses are in geometric series.
Answer to Problem 83E
The total area of all disks is
Explanation of Solution
Formula used:
The formula of sum of infinite geometric series when common factor is smaller than 1 is given by,
Given:
A circular disc is cut out of paper with radius R.
Two disks of radius
Four disks of radius
This process is repeated indefinitely.
Calculation:
The area of a circular disk is,
The area of circular disk of radius R is,
The total area of three circular disks is sum of area of disk of radius R and area of two circular disks of radius
The total area of seven circular disks is sum of area of disk of radius R and area of two circular disks of radius
So, as this process is repeated indefinitely. So the total area of all disks is,
The series from above equation
The first term of series is 1 and the common factor is
So, the sum of the infinite series
So, the total area of all the disks is
Thus, the total area of all the disks is
Chapter 12 Solutions
Precalculus: Mathematics for Calculus - 6th Edition
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