Problem 30 - Precalculus: Functions and Graphs

Question

To measure the height \(h\) of a cloud cover, a meteorology student directs a spotlight vertically upward from the ground. From a point \(P\) on level ground that is \(d\) meters from the spotlight, the angle of elevation \(\theta\) of the light image on the clouds is then measured (see the figure). (a) Express \(h\) in terms of \(d\) and \(\theta\) (b) Approximate \(h\) if \(d=1000 \mathrm{m}\) and \(\theta=59^{\circ}\) (IMAGE CAN NOT COPY)

Step-by-Step Solution

Answer

(a) \(h = d \cdot \tan(\theta)\); (b) \(h \approx 1664.3 \) meters.

Step 1: Set up the problem

We are dealing with a right triangle formed by the spotlight, point P on the ground, and the cloud height. Here, the distance \(d\) is the base of the triangle, and the height \(h\) is the vertical side opposite to the angle \(\theta\).

Step 2: Use the tangent function

In right triangles, the tangent of an angle \(\theta\) is given by the ratio of the opposite side to the adjacent side. Thus, we have:\[\tan(\theta) = \frac{h}{d}\]

Step 3: Solve for height \(h\)

Rearrange the equation to solve for \(h\):\[h = d \cdot \tan(\theta)\]This expresses the height in terms of \(d\) and \(\theta\).

Step 4: Plug in the values

Given \(d = 1000\) meters and \(\theta = 59^{\circ}\), substitute these values into the equation:\[h = 1000 \times \tan(59^{\circ})\]

Step 5: Calculate the height

Use a calculator to find \(\tan(59^{\circ})\), which approximately equals 1.6643 (ensure your calculator is set to degrees, not radians). Therefore:\[h \approx 1000 \times 1.6643 = 1664.3\text{ meters}\]

Key Concepts

Angle of Elevation

The angle of elevation is an important concept in trigonometry and geometry that measures how high one has to look from a certain point to see a distant object. In this context, it's the angle formed between a horizontal line (parallel to the ground) and the line of sight to the top of the object, in this case, the cloud.
This angle is crucial in determining the heights of objects using trigonometry, especially when direct measurement is not feasible.

• It is always measured upwards from the observer's point of view.
• It ranges from 0 degrees, where the object is level with the observer, to 90 degrees, where the object is directly above.

Understanding and measuring the angle of elevation allows us to employ mathematical calculations to find unknown distances or heights using established trigonometric functions.

Right Triangle

A right triangle is one of the simplest geometrical shapes used in trigonometry. It consists of three sides and a right angle, which measures exactly 90 degrees.
In the context of measuring the height of clouds, the right triangle forms naturally during the measurement process.

• The base of the triangle is the horizontal distance from the spotlight's origin to the point directly below the cloud.
• The height of the triangle, or the vertical side, represents the unknown distance you want to measure, which is the cloud height.
• The hypotenuse is the side opposite the right angle, connecting the spotlight to the cloud directly.

The right triangle's properties allow us to implement trigonometric ratios, such as the tangent function, effectively solving for unknown values.

Tangent Function

The tangent function is one of the fundamental trigonometric functions and is very useful in solving right triangle problems. It is specifically related to the angles and sides of a right triangle.
The tangent of an angle \( \theta \) in a right triangle is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. Mathematically, it's expressed as: \[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]

• In our problem, the "opposite" side refers to the cloud height \( h \).
• The "adjacent" side is the distance \( d \) from the spotlight to the reference point on the ground.

By rearranging the equation, we can solve for the unknown height using the measured angle of elevation and the known distance from the spotlight. This conversion is essential for finding unknown heights quickly and accurately.

Cloud Height Measurement

Cloud height measurement using trigonometry is a practical application frequently used in meteorology. Unlike more complex measurement systems that might involve expensive tools or aircraft, using basic trigonometry provides a simple yet effective method for estimating cloud height.
Here's how it typically works:

• A beam of light is directed vertically at the clouds.
• From a known distance on the ground, the angle of elevation is measured using devices like a theodolite.
• The tangent function is then used to determine the cloud height.

This method requires minimal equipment and utilizes basic trigonometric calculations, making it an accessible technique for students and professionals alike. With only a few measurements, you can gather important meteorological data that help understand weather patterns and conditions.