Question 5 of 13intermediate🔧 ApplyNumerical3 marks

Find the number of sides in the nnth step of the shape sequence that leads to the Koch Snowflake.

Correct Answer

Step 1: At Step 0, the equilateral triangle has 3 sides.

Step 2: Each side is replaced by 4 smaller sides in the next step.

Step 3: The number of sides gets multiplied by 4 at each step.

Step 4: Number of sides at step nn = 3×4×4×…3 \times 4 \times 4 \times \dots (n times) = 3×4n3 \times 4^n.

Verification: Step 0: 3×40=33 \times 4^0 = 3 sides Step 1: 3×41=123 \times 4^1 = 12 sides Step 2: 3×42=483 \times 4^2 = 48 sides

Answer: The number of sides at step nn is 3×4n3 \times 4^n.

Exercise: Figure it Out (Section 2) | Q: 2 | (Page Page 4)
For More Understanding

Explanation

The chapter explains that each side of the Koch Snowflake gets replaced by 4 sides at each step (the original side divided into 3 parts, with the middle part replaced by 2 sides of the raised triangle). Starting with 3 sides, each step multiplies the number of sides by 4.

Solution Steps

  1. Step 1: At Step 0, the equilateral triangle has 3 sides.

  2. Step 2: Each side is replaced by 4 smaller sides in the next step.

  3. Step 3: So the number of sides gets multiplied by 4 at each step.

  4. Step 4: Number of sides at step n = 3×43 \times 4 × 4 × ... (n times) = 3 × 4n4^{n}.

  5. Step 5: Verify: Step 0 = 3, Step 1 = 12, Step 2 = 48.

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