Question

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? Answe

Answer 1

The answer would be 6.9 miles

Answer 2

Refer to the attached image.

AB is the distance traveled after taking slightly right onto Oak Avenue from Main street = 4 miles

BC is the distance traveled by taking a 90 degree left turn onto Lilac Lane = x miles.

Lilac Lane intersects Main Street at 30 degrees for the end of the detour, which is AC.

Now we have to find the distance traveled travel on Lilac Lane to reach Main Street = BC=x

Now, in triangle ABC,

by angle sum property,

$\angle A+\angle B+\angle C=180^{\circ}$

$\angle A+90^{\circ}+30^{\circ}=180^{\circ}$

$\angle A=60^{\circ}$

Now in triangle ABC,

Consider $\sin 30^{\circ} = \frac{Perpendicular}{Hypotenuse}$

$\sin 30^{\circ} = \frac{AB}{AC}$

$\frac{1}{2} = \frac{4}{AC}$

AC = 8 miles.

Now consider,

$\sin 60^{\circ}=\frac{BC}{AC}$

$\frac{\sqrt{3}}{2}=\frac{x}{8}$

$\frac{8\sqrt{3}}{2}= x$

$x = 4\sqrt{3}$

x = 6.92 miles

So, 6.92 miles should be traveled on Lilac Lane to reach Main Street.