Question

1. a snow plough with a 3.2 m long blade is set on a 30° angle. what is the width of the path (shown as x in the diagram) that this snow plough will clear?

Explanation:

Step1: Identify the trigonometric relation

We have a right - triangle where the length of the blade is the hypotenuse ($l = 3.2$ m) and the width of the path $x$ is the side adjacent to the given angle $\theta=30^{\circ}$. We use the cosine function $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$.
$\cos\theta=\frac{x}{l}$

Step2: Substitute the values

Substitute $\theta = 30^{\circ}$ and $l = 3.2$ m into the formula. We know that $\cos30^{\circ}=\frac{\sqrt{3}}{2}\approx0.866$.
$x = l\cos\theta$
$x=3.2\times\cos30^{\circ}$
$x = 3.2\times\frac{\sqrt{3}}{2}$
$x = 1.6\sqrt{3}\approx1.6\times1.732 = 2.7712\approx2.77$ m

Answer:

$2.77$ m