4. jason is standing 8.7 km from town x and 11.5 km from town y. from where he stands, the angle between the…

4. 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

4. 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

Answer

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}). [ \begin{align*} XY^{2}&=8.7^{2}+11.5^{2}-2\times8.7\times11.5\times\cos(37^{\circ})\ &=75.69 + 132.25-2\times8.7\times11.5\times0.7986\ &=207.94-(200.1\times0.7986)\ &=207.94 - 159.7\ &=48.24\ XY&=\sqrt{48.24}\approx6.946\space km \end{align*} ]

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

Since the hotel is at the mid - point of (XY), let the mid - point be (M). Then (XM=\frac{XY}{2}\approx\frac{6.946}{2}=3.473\space km)

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) is the distance from Jason to the mid - point of (XY) ((d = 3.473\space km=3473\space m)) [h = d\times\tan(1^{\circ})] [h=3473\times\tan(1^{\circ})] Since (\tan(1^{\circ})\approx0.017455) [h = 3473\times0.017455\approx60.7\space m]

Answer:

The height of the hotel is approximately (60.7\space m)