To find: The total area bounded by the snowflake curve.
Answer to Problem 14P
The perimeter of the snowflake tends to infinity so that the snowflake has no perimeter.
Explanation of Solution
Given information:
The snowflake curve.
Let, the area of snowflake be
Let, the number of sides of snowflake be
Let, the length of the sides of snowflake be
The area of equilateral
The variables above are initialized as follows:
Since the snowflake is an equilateral triangle. It can be easily seen that the length of each side is reduced by a factor three with each iteration.
Therefore,
The number of sides is increased by factor
The perimeter of the snowflake is,
The perimeter of the initial snowflake is,
The perimeter of snowflake can be calculated by G.P. of common ratio. as,
Now taking limit on both sides,
Since the limit tends to infinity for
Therefore, the perimeter of the snowflake tends to infinity so that the snowflake has no perimeter.
Chapter 11 Solutions
Algebra and Trigonometry: Structure and Method, Book 2
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