Question

An orienteer runs for `4 1/2` km, then turns through an angle of 32° and runs for another 6 km. How far is she from her starting point?

Answer 1

`3.233` km (3.d.p.)

Explanation:

If we construct a triangle with side `c= 4.5` side `b=6` and side `a` needs to be found. The angle between `c` and `b` is `32^o`. If we label the triangle so angle `A= 32^o` is opposite the unknown side `a` the we know two sides and the angle between them. For this we can use the Cosine Rule:

`a^2= b^2+c^2-2bc* cos(A)`

Putting in what we know already:

`a^2 = (6)^2+(4.5)^2-2(6)(4.5)*cos(32)`

`a^2 = 36+20.25-54*cos(32)= a^2 = 56.25-54cos(32)`

So:
Taking roots:

`sqrt(a^2)= sqrt(56.25-54cos(32)`

`a= sqrt(56.25-54cos(32))=>a= 3.233` (3.d.p.)