Height Calculation Using Angle of Elevation

This problem involves calculating the height of an object (a hook) on a wall using trigonometry, given the observer's height, distance, and the angle of elevation.

Given Information

• Man's height: $165 \text{ cm}$
• Distance from the wall: $500 \text{ cm}$
• Angle of elevation to the hook: $45^\circ$

Problem Setup

Imagine a right-angled triangle formed by:

• The horizontal line from the man's eye level to the wall (adjacent side).
• The vertical line from the man's eye level up to the hook (opposite side).
• The line of sight from the man's eye to the hook (hypotenuse).

The angle of elevation is the angle at the man's eye level, between the horizontal line and the line of sight to the hook.

Applying Trigonometry

We use the tangent trigonometric function, which is defined as the ratio of the length of the opposite side to the length of the adjacent side in a right-angled triangle:

$ \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} $

In this problem:

• The angle of elevation $\theta = 45^\circ$.
• The adjacent side is the distance from the man to the wall, which is $500 \text{ cm}$.
• The opposite side is the height difference ($h_{diff}$) between the hook and the man's eye level.

Substitute the known values into the formula:

$ \tan(45^\circ) = \frac{h_{diff}}{500 \text{ cm}} $

Since $\tan(45^\circ) = 1$, the equation simplifies to:

$ 1 = \frac{h_{diff}}{500 \text{ cm}} $

Solving for $h_{diff}$:

$ h_{diff} = 1 \times 500 \text{ cm} $

$ h_{diff} = 500 \text{ cm} $

Total Hook Height

The total height of the hook ($H_{hook}$) on the wall is the sum of the man's eye-level height (assumed to be his total height) and the calculated height difference ($h_{diff}$):

$ H_{hook} = \text{Man's height} + h_{diff} $

$ H_{hook} = 165 \text{ cm} + 500 \text{ cm} $

$ H_{hook} = 665 \text{ cm} $

Result

The height at which the hook is located on the wall is 665 cm.