Question

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 town X and town Y

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

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Step2: Find the distance from Jason to the mid - point of \(XY\)

Since the hotel is at the mid - point of \(XY\), let \(d\) be the distance from Jason to the mid - point. Let \(x=\frac{XY}{2}\approx3.475\space km\).
We can use the Law of Cosines again in the new triangle (with sides \(8.7\), \(11.5\), and \(XY\)). Another way is to use the formula for the length of the median. The formula for the length of the median \(m\) from a vertex to the mid - point of the opposite side in a triangle with sides \(a\), \(b\), \(c\) is \(m=\frac{1}{2}\sqrt{2a^{2}+2b^{2}-c^{2}}\).

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Step3: Use the tangent function to find the height of the hotel

We know that \(\tan\theta=\frac{h}{d}\), where \(\theta = 1^{\circ}\) and \(d = 9.585\space km=9585\space m\)

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

$$h=9585\times0.017455$$

$$h\approx167.3\space m$$

Answer:

The height of the hotel is approximately \(167.3\space m\)