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 sine formula

The sine formula is \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\). Here, the angle of elevation is \(1^{\circ}\), and we need to find the height \(h\) of the hotel. The hypotenuse is the distance from Jason to the point on the road - hotel connection (let's assume the line - of - sight from Jason to the top of the hotel).

Step2: Calculate the height

We know that \(\sin(1^{\circ})=\frac{h}{10}\) (assuming the length of the line - of - sight from Jason to the top of the hotel is \(10\) km. This is an assumption based on the problem's geometric setup. If we consider the right - triangle formed by the height of the hotel (\(h\)), the line - of - sight (let's call it \(l\)) and the horizontal distance. From the problem's geometric interpretation, if we assume the non - height side of the right - triangle (related to the angle of elevation) is \(l = 10\) km.

$$h = 10\times\sin(1^{\circ})$$

We know that \(\sin(1^{\circ})\approx0.0174524\)

$$h=10\times0.0174524 = 0.174524\space km$$

Since \(1\space km = 1000\space m\), then \(h=0.174524\times1000=174.524\approx175\space m\)

Answer:

The height of the hotel is approximately \(175\) metres.