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.