find the value of $x$ and $y$.

Explanation:

Step1: Identify the trigonometric relationship for $x$

Using the tangent ratio for the $30^{\circ}$ angle: $$\tan(30^{\circ}) = \frac{\text{opposite}}{\text{adjacent}} = \frac{x}{21}$$

Step2: Solve for $x$

Multiply both sides by 21 and substitute $\tan(30^{\circ}) = \frac{\sqrt{3}}{3}$: $$x = 21 \cdot \frac{\sqrt{3}}{3} = 7\sqrt{3}$$

Step3: Identify the trigonometric relationship for $y$

Using the cosine ratio for the $30^{\circ}$ angle: $$\cos(30^{\circ}) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{21}{y}$$

Step4: Solve for $y$

Rearrange the equation and substitute $\cos(30^{\circ}) = \frac{\sqrt{3}}{2}$: $$y = \frac{21}{\cos(30^{\circ})} = \frac{21}{\frac{\sqrt{3}}{2}} = \frac{42}{\sqrt{3}} = 14\sqrt{3}$$

Answer:

$x = 7\sqrt{3}$ $y = 14\sqrt{3}$