Question

find the length x to the nearest whole number.
340
60°
30°
x

Explanation:

Step1: Use tangent function for inner - triangle

Let the length of the part of the vertical side below the smaller right - triangle be \(y\). For the larger right - triangle with an angle of \(30^{\circ}\), \(\tan30^{\circ}=\frac{x}{340}\), so \(x = 340\tan30^{\circ}\). For the smaller right - triangle with an angle of \(60^{\circ}\), if we assume the vertical side of the smaller right - triangle is \(h\), \(\tan60^{\circ}=\frac{x}{h}\), \(h=\frac{x}{\tan60^{\circ}}\). Also, \(340 - h\) is the vertical side of the non - overlapping part of the larger right - triangle. But we can also use the fact that for the larger right - triangle with angle \(\theta = 30^{\circ}\) and opposite side \(x\) and adjacent side \(340\), and for the smaller right - triangle with angle \(\alpha=60^{\circ}\) and adjacent side \(a\) and opposite side \(x\).
We know that \(\tan30^{\circ}=\frac{x}{340}\), and \(\tan30^{\circ}=\frac{1}{\sqrt{3}}\).

Step2: Calculate the value of \(x\)

\[x = 340\times\frac{1}{\sqrt{3}}\approx340\times0.577 = 196.18\approx196\]

Answer:

196