Due to construction, traffic is getting detoured from Main Street by turning slightly right onto Oak Avenue and continuing straight for 4 mi. To get back onto Main Street, you need to make a 90 degree left turn onto Lilac Lane and continue straight for x miles. Lilac Lane intersects Main Street at 30 degrees for the end of the detour. Approximately how many miles should you travel on Lilac Lane to reach Main Street?

6.9 miles

Assuming that Main Street, Oak Avenue, and Lilac Lane are all straight, you have a right 30/60/90 right triangle. I do object to the phrase "turning slightly right onto Oak" since I wouldn't consider a 60 degree turn to be "slight". But in any case, the short leg of the 30/60/90 triangle is the 4 miles given and the long leg will be sqrt (3) times longer. So

sqrt (3) * 4 miles = 1.732050808 * 4 miles = 6.92820323 miles

So you would need to travel about 6.9 miles on Lilac Lane to reach Main Street.

For this case you should see the problem as a rectangle triangle whose sides are 4 miles (base) and x miles (height).

the angle between the hypotenuse and the base is 30 degrees.

then to find the height:

tan (30) = x / 4

x = 4 * tan (30) = 2.31

answer

you should travel 2.31 miles on Lilac Lane to reach Main Street