https://r2.erweima.ai/mathbot/image/17265144622135v0itgx4.jpg

The exercise asks to "Calculate the length of each curve" for the given segments of circles. Each curve corresponds to an arc, and to find the length of an arc, the formula used is:

\[
L = 2\pi r \cdot \frac{\theta}{360^\circ}
\]

Where:
- \( L \) is the arc length,
- \( r \) is the radius of the circle,
- \( \theta \) is the angle of the arc in degrees,
- \( \pi \) is approximated as 3.14.

Let's break down each part from the image:

### a. Arc with radius \( r = 12 \, \text{cm} \)
- Angle not explicitly given, but likely needs clarification or typical value to proceed.

### b. Arc with radius \( r = 20 \, \text{cm} \)
- Angle not given, needs clarification or assumed value.

### c. Arc with radius \( r = 8 \, \text{cm} \)
- Again, the angle is not provided. This needs further clarification.

### d. Arc with radius \( r = 15 \, \text{cm} \) and angle \( \theta = 126^\circ \)

Let's start by calculating the length of the arc for part **d**, where the values are complete:

#### **d. Arc Length Calculation**

- \( r = 15 \, \text{cm} \),
- \( \theta = 126^\circ \).

\[
L = 2 \pi r \cdot \frac{\theta}{360^\circ} = 2 \cdot 3.14 \cdot 15 \cdot \frac{126}{360}
\]

I will now calculate the arc length for part **d**.The length of the arc for part **d** is approximately \( 32.97 \, \text{cm} \).

Let me know if you need further calculations or clarification. 

Here are some related questions:
1. How would the arc length change if the radius were doubled in part d?
2. How can you calculate the full circumference of a circle using the radius?
3. What formula would you use if you were asked to find the area of a sector instead of the arc length?
4. What is the relationship between the angle of the arc and the fraction of the circle it represents?
5. How does the approximation of \( \pi = 3.14 \) affect your calculations compared to using \( \pi \approx 3.1416 \)?

**Tip:** Always make sure the angle \( \theta \) is measured in degrees for the arc length formula.

Calcula la longitud de cada curva (Ejercicio 22) basado en el cálculo de arcos de circunferencia.

Learn how to calculate the length of arcs for circle segments in different exercises. Using the formula L = 2πr * (θ / 360), explore the geometric properties of arcs and their relation to the radius and angle. Suitable for Grades 9-12 students.

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