Problem Analysis: Hotel Heights and Angle of Depression

The question describes a scenario involving two hotels of different heights standing a certain distance apart. We are given the height of one hotel, the distance between the hotels, and the angle of depression from the top of the taller hotel to the top of the shorter hotel. We need to determine the height of the second hotel.

Understanding Angle of Depression

The angle of depression is the angle between the horizontal line from the observer's eye to an object below the horizontal line. In this problem, the observer is at the top of the taller hotel, and the object is the top of the shorter hotel.

Imagine a horizontal line extending from the top of the taller hotel. The angle measured downwards from this horizontal line to the line of sight connecting the tops of the two hotels is the angle of depression. This angle is equal to the alternate interior angle formed by the line of sight and a horizontal line extending from the top of the shorter hotel.

Setting up the Geometry

Let's represent the situation geometrically.

• Let the height of the first hotel (the taller one) be \(h_1 = 70\) m.
• Let the height of the second hotel be \(h_2\). This is what we need to find.
• The distance between the two hotels is \(d = 25\) m.

Consider the tops of the two hotels, let's call them T1 (taller) and T2 (shorter). Let the bases be B1 and B2 respectively. So, T1B1 = \(h_1 = 70\) m, T2B2 = \(h_2\), and B1B2 = \(d = 25\) m.

Draw a horizontal line from T1 that extends towards the second hotel. Let this line intersect the vertical line passing through T2 at point P. Then, T1P is parallel and equal to B1B2, so T1P = 25 m. The line segment T2P represents the vertical distance between the top of the shorter hotel (T2) and the horizontal line from the top of the taller hotel (T1).

This forms a right-angled triangle \(\triangle T1PT2\), where the right angle is at P.

• The angle of depression from T1 to T2 is 45°. This is the angle between the horizontal line T1P and the line T1T2. So, \(\angle PT1T2 = 45^\circ\).
• The side T1P is the horizontal distance between the tops, which is equal to the distance between the bases: T1P = 25 m.
• The side T2P is the vertical distance between the tops. T2P = T1B1 - T2B2 = \(h_1 - h_2\).

Calculations using Trigonometry

In the right-angled triangle \(\triangle T1PT2\), we know the angle \(\angle PT1T2 = 45^\circ\) and the adjacent side T1P = 25 m. We want to find the opposite side T2P.

The trigonometric ratio that relates the opposite side and the adjacent side in a right triangle is the tangent function.

We have:

\(\tan(\angle PT1T2) = \frac{\text{Opposite side}}{\text{Adjacent side}}\)

\(\tan(45^\circ) = \frac{\text{T2P}}{\text{T1P}}\)

We know that \(\tan(45^\circ) = 1\). Substituting the known values:

\(1 = \frac{\text{T2P}}{25}\)

Now, we can solve for T2P:

\(\text{T2P} = 1 \times 25\)

\(\text{T2P} = 25\) m

The vertical distance T2P is the difference in height between the two hotels:

\(\text{T2P} = h_1 - h_2\)

\(25 = 70 - h_2\)

To find the height of the second hotel, \(h_2\), we rearrange the equation:

\(h_2 = 70 - 25\)

\(h_2 = 45\) m

Final Answer

The height of the other hotel is 45 m.

Revision Table: Key Concepts in Trigonometry Application

Additional Information: Applications of Trigonometry

Trigonometry is a branch of mathematics dealing with the relationships between the sides and angles of triangles, especially right-angled triangles. It has numerous applications in various fields.

• Surveying and Cartography: Used to measure distances, heights, and angles in the real world to create maps and surveys. Measuring the height of buildings, mountains, or the distance across rivers often involves trigonometric principles.
• Navigation: Essential in celestial navigation (using stars and planets), GPS systems, and aircraft navigation to calculate positions and directions.
• Physics and Engineering: Used extensively in mechanics, optics, acoustics, and electrical engineering to model periodic phenomena (like waves) and analyze forces and motions.
• Astronomy: Used to calculate the distances between celestial bodies and study their movements.
• Architecture: Helps in designing structures, calculating roof slopes, building heights, and ensuring stability.

Problems involving angles of elevation and depression are classic examples of applying trigonometry to solve real-world height and distance calculations, just like finding the height of the hotel in this question.