Question

question 9 (mandatory) (1 point) determine the length of x, to the nearest tenth of a metre. a) 123.4 m b) 202.2 m c) 115.6 m d) 103.0 m

Explanation:

Step1: Use the Law of Sines in the lower - triangle

The Law of Sines states that \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\). Let's first find the length of the side adjacent to the \(27^{\circ}\) angle in the right - triangle (where \(x\) is the height).
In the non - right triangle with side \(225\) m:
Let the angles be \(A = 51^{\circ}\), \(B=65^{\circ}\), \(C = 180-(51 + 65)=64^{\circ}\).
By the Law of Sines \(\frac{a}{\sin A}=\frac{225}{\sin C}\). Let \(a\) be the side we want to find (adjacent to the \(27^{\circ}\) angle in the right - triangle). So \(a=\frac{225\times\sin51^{\circ}}{\sin64^{\circ}}\).
We know that \(\sin51^{\circ}\approx0.777\), \(\sin64^{\circ}\approx0.899\). Then \(a=\frac{225\times0.777}{0.899}=\frac{174.825}{0.899}\approx194.5\) m.

Step2: Use the right - triangle trigonometry

In the right - triangle (with angle \(27^{\circ}\) and the side found in Step 1 as the adjacent side to the \(27^{\circ}\) angle and \(x\) as the opposite side).
We know that \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). Here \(\theta = 27^{\circ}\), and the adjacent side \(a\approx194.5\) m.
Since \(\tan27^{\circ}\approx0.5095\), and \(\tan27^{\circ}=\frac{x}{a}\), then \(x=a\times\tan27^{\circ}\).
Substitute \(a\approx194.5\) into the formula: \(x = 194.5\times0.5095\approx103.0\) m.

Answer:

d) \(103.0\) m