Question

trigonometry assignment t /29

1. jason is standing 8.7 km from town x and 11.5 km from town y. from where he

stands, the angle between the two towns is 37°. a new hotel has just been built on
the road connecting town x and town y, exactly halfway between the two towns.
from where jason is standing, he sees that the angle of elevation to the top of the
hotel is 1°. determine the height of the hotel, to the nearest tenth of a metre. 7t

Explanation:

Step1: Use the Law of Cosines to find the distance between \(X\) and \(Y\)

The Law of Cosines formula is \(c^{2}=a^{2}+b^{2}-2ab\cos C\). Let \(a = 8.7\), \(b=11.5\), and \(C = 37^{\circ}\).

$$
LATEXBLOCK0
$$

The distance from the mid - point of \(XY\) to Jason: Let \(d\) be the distance from the mid - point of \(XY\) to Jason. Since the mid - point of \(XY\) (length \(l = XY\approx6.946\space km\)) divides \(XY\) into two equal parts of length \(l/2\approx3.473\space km\).
We can also use the formula \(d=\frac{2ab\cos\frac{C}{2}}{a + b}\) (another form for the distance from a point to the mid - point of the side opposite in a triangle). But using the Law of Cosines in the sub - triangle formed by half of \(XY\), \(8.7\), and \(d\) (or \(11.5\), half of \(XY\) and \(d\)):
Let \(x=\frac{XY}{2}\approx3.473\space km\), \(a = 8.7\), \(C = 37^{\circ}\)

$$
LATEXBLOCK1
$$

Step2: Use the tangent function to find the height of the hotel

We know that \(\tan\theta=\frac{h}{d}\), where \(\theta = 1^{\circ}\) and \(d\approx5517\space m\)

$$h = d\times\tan(1^{\circ})$$

$$h=5517\times0.017455\approx96.3\space m$$

Answer:

\(96.3\space m\)