In the Golden ratio spiral, What is the proportion of the arc lengths of the orange 13x13 
square and pink 21x21 square? Copy problem; show solution 
on paper 

To solve this problem, we are asked to find the proportion of the arc lengths in two squares that are part of a Golden Ratio spiral. The orange square has dimensions \( 13 \times 13 \) and the pink square has dimensions \( 21 \times 21 \). The arc length of each square corresponds to a portion of the Fibonacci spiral, where the arc is a quarter-circle inscribed in each square.

The formula for the arc length of a quarter circle is given by:

\[
\text{Arc length} = \frac{1}{4} \times 2\pi r = \frac{\pi r}{2}
\]

where \( r \) is the radius of the arc, which is the same as the side length of the square.

### Step-by-step solution:

1. **Arc length of the orange square (13x13):**
   \[
   r_{\text{orange}} = 13
   \]
   The arc length of the quarter-circle in the orange square:
   \[
   L_{\text{orange}} = \frac{\pi \times 13}{2} = \frac{13\pi}{2}
   \]

2. **Arc length of the pink square (21x21):**
   \[
   r_{\text{pink}} = 21
   \]
   The arc length of the quarter-circle in the pink square:
   \[
   L_{\text{pink}} = \frac{\pi \times 21}{2} = \frac{21\pi}{2}
   \]

3. **Proportion of the arc lengths:**
   The proportion of the arc length of the orange square to the pink square is:
   \[
   \frac{L_{\text{orange}}}{L_{\text{pink}}} = \frac{\frac{13\pi}{2}}{\frac{21\pi}{2}} = \frac{13}{21} = \frac{13}{21} = \frac{13}{21}
   \]
   Simplifying this ratio, we get:
   \[
   \frac{13}{21}
   \]

Therefore, the proportion of the arc lengths between the orange square and the pink square is \( \frac{13}{21} \).

---

Let me know if you'd like a handwritten version of the solution! Would you also like more details, or do you have any questions about this? Here are five related questions for further exploration:

1. What is the Fibonacci sequence, and how is it related to the Golden Ratio?
2. How can the Golden Ratio be calculated, and why is it important in geometry?
3. Can you derive the total arc length of the entire Golden Ratio spiral up to a given Fibonacci number?
4. What is the connection between the Golden Ratio and natural phenomena, such as shells and flowers?
5. How does the geometry of a logarithmic spiral differ from that of an Archimedean spiral?

**Tip:** The Golden Ratio is approximately \( 1.618 \) and is denoted by \( \phi \). It can be calculated from the formula \( \phi = \frac{1 + \sqrt{5}}{2} \).

In the Golden ratio spiral, What is the proportion of the arc lengths of the orange 13x13 square and pink 21x21 square? Copy problem; show solution on paper

Explore the calculation of the proportion of arc lengths in a Golden Ratio spiral involving 13x13 and 21x21 squares. Understand the connection between geometry, the Fibonacci sequence, and the Golden Ratio with detailed step-by-step solutions. Ideal for high school students studying advanced geometry.

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