Question

At the foot of a mountain the elevation of its peak is found to be \(\frac{\pi}{4}\). After ascending 'h' m toward the mountain up a slope of \(\frac{\pi}{6}\) inclination, the elevation is found to be \(\frac{\pi}{3}\). Height of the mountain is (1) \(\frac{h}{2}(\sqrt{3}+1) m\) (2) \(\mathrm{h}(\sqrt{3}+1) \mathrm{m}\) (3) \(\frac{\mathrm{h}}{2}(\sqrt{3}-1) \mathrm{m}\) (4) \(\mathrm{h}(\sqrt{3}-1) \mathrm{m}\)

Short answer

\( \frac{h}{2}(\sqrt{3} + 1) \)

Step-by-step solution

- Identify the given angles

At the foot of the mountain, the elevation angle of the peak is given as \(\frac{\pi}{4}\). After moving 'h' meters up a slope inclined at \(\frac{\pi}{6}\), the new elevation angle is \(\frac{\pi}{3}\).

- Relate the elevation change to the slope

Using trigonometric principles, determine the vertical height gained (let's call it 'a') after moving 'h' meters along the slope. This can be given by the equation: \[ a = h \sin \left(\frac{\pi}{6}\right) = \frac{h}{2} \]

- Calculate the height using trigonometric relationships

Setup the relationship using right triangles formed by the ground, the height of the peak, and the observer points. Let H be the total height of the mountain. Initially, \(\tan\left(\frac{\pi}{4}\right) = \frac{H}{x}\) where x is the horizontal distance to the peak. Therefore, \[ H = x \].

- Establish equation after ascent

After ascending, the new horizontal distance to the peak changes to \(x - \frac{h}{2}\), and the new height from this point forms another right triangle: \[ \tan\left(\frac{\pi}{3}\right) = \frac{H - \frac{h}{2}}{x - \frac{h}{2}} \] Simplify using \(\tan\left(\frac{\pi}{3}\right) = \sqrt{3}\) to give: \[ \sqrt{3}(x - \frac{h}{2}) = H - \frac{h}{2} \]

- Solve for H

Substitute \(H = x\) from the initial relationship: \[ \sqrt{3}x - \frac{\sqrt{3}h}{2} = x - \frac{h}{2} \] Rearrange to solve for x: \[ \sqrt{3}x - x = \frac{h}{2}(\sqrt{3} - 1) \] \[ x(\sqrt{3} - 1) = \frac{h}{2}(\sqrt{3} - 1) \] \[ x = \frac{h}{2} \] Therefore, the height H is: \[ H = x = \frac{h}{2}(\sqrt{3} + 1) \]

Key concepts

elevation angle

In trigonometry, the **elevation angle** is the angle formed by the line of sight above the horizontal. Imagine you are looking at the top of a mountain from a certain point. The angle your gaze makes with the ground is the elevation angle. The problem gives that at the foot of the mountain, this angle is \( \frac{\pi}{4} \).

When you move a certain distance \( h \) towards the mountain along a slope, the elevation angle changes to \( \frac{\pi}{3} \). These elevation angles help us understand how high the peak of the mountain is from different vantage points. Elevation angles are crucial in solving height problems using trigonometric relationships. They allow us to use the properties of right triangles to find unknown heights.

trigonometric relationships

Trigonometric relationships are mathematical connections between the angles and sides of triangles, particularly right triangles.

For this problem, we utilize the tangent function, defined as:

• \( \text{Tan(Angle)} = \frac{\text{Opposite Side}}{\text{Adjacent Side}} \).

Initially, we have \( \tan\left( \frac{\pi}{4} \right) = \frac{H}{x} \), where \( H \) is the height of the mountain and \( x \) is the horizontal distance from the observer to the mountain's base.

Since \( \tan\left( \frac{\pi}{4} \right) = 1 \), we conclude that \( H = x \).

After moving\( h \) meters up a slope inclined at \( \frac{\pi}{6} \), the problem provides a new elevation angle of \( \frac{\pi}{3} \).

To solve this using trigonometric relationships, consider that we have shifted our position horizontally by \( \frac{h}{2} \) meters (calculated using \( h \cos(\frac{\pi}{6}) \)). From the new position, the equation transforms to \( \tan(\frac{\pi}{3}) = \frac{H - \frac{h}{2}}{x - \frac{h}{2}} \), resulting in a system of equations that help solve for the mountain's height.

right triangle analysis

Right triangle analysis involves the study of triangles with one 90-degree angle. This concept is essential because it allows us to apply trigonometric ratios like sine, cosine, and tangent.

For this problem, imagine the initial observer point, the top of the mountain, and the peak forming a right triangle, where the tangent relationship initially provided us with \( H = x \).

After moving up the slope, the right triangle's configuration changes. We determine the vertical height gained as \( a = h \sin(\frac{\pi}{6}) = \frac{h}{2} \).

A new right triangle forms between the new observer point, the horizontal component of the new position (\( x - \frac{h}{2} \)), and the remaining height of the mountain (\( H - \frac{h}{2} \)).

We use \( \tan(\frac{\pi}{3}) \) to set up the relationship \( \sqrt{3}(x - \frac{h}{2}) = H - \frac{h}{2} \). By substituting \( x = H \), it simplifies to solve for \( H = \frac{h}{2}(\sqrt{3} + 1) \), the height we sought initially.

This analysis leverages the fundamental relationships within right triangles to derive critical measurements.