Chapter 12, Problem 1E

Consider the construction of a Koch snowflake starting with a seed triangle having sides of length 81 cm. Let M denote the number of sides, L the length of each side, and P the perimeter of the "snowflake" obtained at the indicated step of the construction. Complete the missing entries in Table 12-1 Q.

Table 12-1

	M	L	P
Start	3	81 cm	243 cm
Step 1	12	27 cm	324 cm
Step 2
Step 3
Step 4
Step 5

To determine

To complete:

The given chart.

Answer to Problem 1E

Solution:

The given chart is,

	M	L	P
Start	3	$81\text{cm}$	$243\text{cm}$
Step 1	12	$27\text{cm}$	$324\text{cm}$
Step 2	48	$9\text{cm}$	$432\text{cm}$
Step 3	192	$3\text{cm}$	$576\text{cm}$
Step 4	768	$1\text{cm}$	$768\text{cm}$
Step 5	3072	$\frac{1}{3}\text{cm}$	$1024\text{cm}$

Explanation of Solution

Given:

The given chart is,

	M	L	P
Start	3	81 cm	243 cm
Step 1	12	27 cm	324 cm
Step 2
Step 3
Step 4
Step 5

Calculation:

Draw an equilateral triangle with side length of $81\text{cm}$. The below figure shows the equilateral triangle.

Figure-1

To the middle third of each of the sides of this seed, add an equilateral triangle with sides of length 27 cm. The result is a 12 sided snowflake with perimeter,

$12\times 27=342\text{cm}$

Figure-2

To the middle third of each of the 12 sides of snowflake add an equilateral triangle with sides of length one third of the length of the side. The result is a snowflake with 48 sides each side length of 9 cm and the perimeter is,

$48\times 9=432$

Figure-3

	M	L	P
Start	3	$81\text{cm}$	$243\text{cm}$
Step 1	12	$27\text{cm}$	$324\text{cm}$
Step 2	48	$9\text{cm}$	$432\text{cm}$
Step 3	192	$3\text{cm}$	$576\text{cm}$
Step 4	768	$1\text{cm}$	$768\text{cm}$
Step 5	3072	$\frac{1}{3}\text{cm}$	$1024\text{cm}$

Therefore, the given chart calculations are stated above.