Question

Central Boulevard runs diagonally across town as shown below. The distance from Eastern Avenue to Western Avenue along South Street is 4(sqrt)3 miles.

Answer 1

The locations of the streets describes a right triangle with the angles 30 degrees and 60 degrees between them and the length of the side opposite the 60 degrees angle is $4\sqrt{3} \, miles$

a.) To find the distance form North Street to South Street along Eastern Avenue, we find the length of the side opposite the 30 degrees angle.

Using sine rule, we recall that the sine rule states that
$\frac{\sin{A}}{a} = \frac{\sin{B}}{b}$
where:
a = $4\sqrt{3}$
A = 60 degrees
B = 30 degrees
b = unknown
$\frac{\sin{60^o}}{4\sqrt{3}} = \frac{\sin{30^o}}{b} \\ b=\frac{4\sqrt{3} \sin{30^o}}{\sin{60^o}}= \frac{3.4641}{0.8660} = 4 \, miles$

Therefore, the distance form North Street to South Street along Eastern Avenue is 4 miles.

b.) To find the distance from North Street to South Street along Central Boulevard we find the length of the hypothenus of the right triangle.

Using pythagoras theorem, we recall that the pythagoras states that
$c^2=a^2+b^2$
where:
a = $4\sqrt{3}$
b = 4 miles
c = hypothenus = unknown
$c^2=\left(4 \sqrt{3} \right)^2+4^2=48+16=64 \\ c= \sqrt{64} =8 \, miles$

Therefore, the distance from North Street to South Street along Central Boulevard is 8 miles