A curve on a railroad track consists of two circular arcs that make an S shape. The central angle of one is 20° with radius 2500 ft and the central angle of the other is 25° with radius 3000 ft. Find the total length of the two arcs.

To find the total length of the two arcs, we'll use the formula for the length of an arc of a circle:

\[
\text{Arc Length} = r \times \theta
\]

where:
- \( r \) is the radius of the arc,
- \( \theta \) is the central angle in radians.

Since the central angles are given in degrees, we first need to convert them into radians. The conversion from degrees to radians is:

\[
\text{Radians} = \text{Degrees} \times \frac{\pi}{180}
\]

### Step 1: Convert the central angles to radians

For the first arc:
\[
\theta_1 = 20^\circ \times \frac{\pi}{180} = \frac{20\pi}{180} = \frac{\pi}{9} \text{ radians}
\]

For the second arc:
\[
\theta_2 = 25^\circ \times \frac{\pi}{180} = \frac{25\pi}{180} = \frac{5\pi}{36} \text{ radians}
\]

### Step 2: Calculate the length of each arc

For the first arc:
\[
\text{Length of the first arc} = r_1 \times \theta_1 = 2500 \times \frac{\pi}{9} \approx 872.66 \text{ ft}
\]

For the second arc:
\[
\text{Length of the second arc} = r_2 \times \theta_2 = 3000 \times \frac{5\pi}{36} \approx 1308.99 \text{ ft}
\]

### Step 3: Find the total length of the two arcs

\[
\text{Total length} = 872.66 \text{ ft} + 1308.99 \text{ ft} = 2181.65 \text{ ft}
\]

### Final Answer

The total length of the two arcs is approximately **2181.65 feet**.

Would you like more details or have any questions?

Here are 5 related questions to consider:
1. How would the total length change if the central angles were doubled?
2. What is the length of an arc if the central angle is 45° with a radius of 2000 ft?
3. How do you find the area of a sector with a central angle of 30° and a radius of 2500 ft?
4. What is the relationship between arc length and chord length for a given radius?
5. How do you calculate the radius of an arc if the length and central angle are known?

**Tip:** Always make sure to convert angles to radians when working with arc lengths in formulas.

Learn how to calculate the total length of two S-shaped railroad track arcs, each forming part of a circular arc. This problem involves geometry and the calculation of arc lengths using central angles and radii. Suitable for high school students.

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