Problem 60 - Algebra and Trigonometry

Question

Height of Cloud Cover To measure the height of the cloud cover at an airport, a worker shines a spotlight upward at an angle \(75^{\circ}\) from the horizontal. An observer 600 \(\mathrm{m}\) away measures the angle of elevation to the spot of light to be \(45^{\circ} .\) Find the height \(h\) of the cloud cover.

Step-by-Step Solution

Answer

The height of the cloud cover is 600 meters.

Step 1: Identify the Triangle Components

First, identify the components of the right triangle formed by the worker, the observer, and the cloud cover. The observer is 600 m away from the point directly below the light, and the angles provided are \(75^{\circ}\) and \(45^{\circ}\). The height \(h\) of the cloud cover is represented as the vertical side opposite the \(45^{\circ}\) angle in the triangle.

Step 2: Use Trigonometric Ratios

Since the observer measures the angle of elevation to the spot of light as \(45^{\circ}\), use the tangent function to relate the height of the cloud cover to the observer's distance. \(\tan(45^{\circ}) = \frac{h}{600}\). Recall that \(\tan(45^{\circ}) = 1\).

Step 3: Solve for Height h

Since \(\tan(45^{\circ}) = 1\), the equation simplifies to \(1 = \frac{h}{600}\). Multiply both sides by 600 to solve for \(h\), giving \(h = 600\).

Step 4: Verify the Calculation

Upon solving using \(\tan(45^{\circ}) = 1\), we find \(h = 600\). Reiterating the setup confirms that the vertical component measured by the observer is indeed correctly translated into the \(600 \text{ meters}\) of height, as determined from the spot on the cloud to the observer on the ground.

Key Concepts

angles

Angles are fundamental in many areas of mathematics, especially in trigonometry, where they help us understand relationships in triangles. An angle is formed by two rays meeting at a common endpoint, called the vertex.
In this exercise, we encounter two specific angles - one at \(75^{\circ}\) and another at \(45^{\circ}\). These angles are crucial because they help determine the height of the cloud cover..

• The \(75^{\circ}\) angle is the angle between the spotlight beam and the horizontal ground, originating from the airport's worker.
• The \(45^{\circ}\) angle is the angle of elevation measured by the observer who is located 600 meters away.

Understanding these angles allows you to determine the relative positions and measurements necessary for calculating the height using trigonometric functions.

right triangle

The right triangle is a special type of triangle in which one of the angles measures \(90^{\circ}\). The right triangle in this problem is formed by: the line from the observer to the point directly below the spot of light and the vertical line (height of the cloud cover) from this point directly to the cloud.

Important components of a right triangle include:

• The hypotenuse - the longest side, opposite the \(90^{\circ}\) angle.
• The opposite side - the side opposite the angle in question, in this case, the height \(h\).
• The adjacent side - the side forming the angle with the angle in question, here, 600 meters along the ground.

In our scenario, we leverage these components to use trigonometric properties, helping find unknown measurements, such as the height, using known lengths and angles.

tangent function

The tangent function is one of the primary functions in trigonometry and is especially useful in problems involving right triangles. The tangent of an angle is the ratio of the opposite side to the adjacent side in a right triangle.
For an angle \(\theta\), the tangent function is defined as:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]
In this problem, we specifically used \(\tan(45^{\circ})\). Interestingly, tangent is sometimes uniquely easy to work with, especially if one of the angles is \(45^{\circ}\). For \(45^{\circ}\), the tangent value is 1, meaning the opposite and adjacent sides are equal.

• For this problem, applying \(\tan(45^{\circ}) = 1\) gave us the equation \(1 = \frac{h}{600}\).
• This simplified the solution, leading directly to finding that \(h = 600\).

The efficient use of the tangent function simplifies finding heights and distances in real-world applications.