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: Identify the trigonometric relationship

We know that in a right - triangle, \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\). Here, the angle of elevation \(\theta = 37^{\circ}\) and the hypotenuse (the distance from Jason to the top of the hotel) is \(10\) km. Let the height of the hotel be \(h\).

Step2: Apply the sine formula

Using the formula \(\sin\theta=\frac{h}{10}\) (where \(\theta = 37^{\circ}\)), and we know that \(\sin(37^{\circ})\approx0.6018\).
Substitute \(\sin(37^{\circ})\) into the formula: \(h = 10\times\sin(37^{\circ})\)

Step3: Calculate the value

\(h=10\times0.6018 = 6.018\approx6.0\) km

Answer:

The height of the hotel is approximately \(6.0\) km.