To solve this problem, we need to calculate the length of the outer curve of the road. Since the path is part of a circle, we can use the formula for the length of an arc:

$\text{Arc Length} = r \cdot \theta$

where:

• $r$ is the radius of the circle,
• $\theta$ is the central angle in radians.

Step 1: Determine the radii

• The radius of the inner curve is $r_{\text{inner}} = 50$ feet.
• The width of the road is 20 feet, so the radius of the outer curve is $r_{\text{outer}} = 50 + 20 = 70$ feet.

Step 2: Convert the angle from degrees to radians

• The central angle $\theta$ is given as 70 degrees.
• To convert degrees to radians:

$\theta_{\text{radians}} = \theta_{\text{degrees}} \times \frac{\pi}{180} = 70 \times \frac{3.14}{180} = 1.22 \text{ radians (approx)}$

Step 3: Calculate the length of the outer curve

Using the outer radius $r_{\text{outer}} = 70$ feet and the angle in radians $\theta = 1.22$ radians:

$\text{Arc Length} = r_{\text{outer}} \times \theta = 70 \times 1.22 = 85.4 \text{ feet}$

Step 4: Round to the nearest foot

The nearest whole number is 85 feet, which is closest to 86 feet.

Final Answer

The length of the outer curve is approximately 86 feet, which corresponds to option C.

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Would you like further details or have any questions? Here are some related questions to consider:

1. How do you calculate the arc length for different angles?
2. What are some practical applications of calculating arc lengths in real-world scenarios?
3. How do you convert angles between degrees and radians?
4. What is the relationship between the radius and arc length of a circle?
5. How would the arc length change if the road width was increased?

Tip: Always ensure the angle is in radians when using the formula $\text{Arc Length} = r \cdot \theta$.