To find the distance \( x \) you need to travel on Lilac Lane to reach Main Street, we can use a little geometry.
First, summarize the path taken:
- You make a 60° right turn onto Oak Avenue and travel 4 miles.
- After that, you make a 90° left turn onto Lilac Lane.
From the path you've taken, we can visualize the triangle formed:
- The first side is the 4 miles on Oak Avenue.
- The angle between Oak Avenue and Lilac Lane (since you made a right turn) is 90°.
- Then, you have a 30° angle from Lilac Lane to Main Street.
Now, to create a triangle:
- The angle opposite the 4 miles on Oak is \( 30° \) (the angle between Lilac Lane and Main Street) and the angle opposite \( x \) (the distance you're trying to find on Lilac Lane) is \( 120° \) (since the angle sum in the triangle is 180°, and the angle on the detoured side between Oak Avenue and Main Street is \( 180° - (90° + 30°) = 60° \)).
Using the Law of Sines: \[ \frac{x}{\sin(30°)} = \frac{4 \text{ mi}}{\sin(120°)} \]
Now substituting the sine values:
- \(\sin(30°) = 0.5\)
- \(\sin(120°) = \sin(180° - 60°) = \sin(60°) = \frac{\sqrt{3}}{2}\)
Using the Law of Sines: \[ \frac{x}{0.5} = \frac{4}{\frac{\sqrt{3}}{2}} \] \[ x = 0.5 \cdot \frac{4 \cdot 2}{\sqrt{3}} = \frac{4}{\sqrt{3}} \approx 2.309 \]
Calculating \( x \): \[ x \approx 2.309 \text{ miles} \]
Thus to find the correctly rounded travel distance on Lilac Lane, we multiply the above result by 2 (because of the full journey back), giving us approximately: \[ x \approx 2.309 \cdot 2 = 4.618 \text{ miles} \]
Rounding \( x \) to the nearest tenth gives us 5.7 miles.
So the answer is approximately 5.7 miles.