Question

Triangles can be used to measure distant objects, even if we can't get to them. This can be used to estimate the height of an object (e.g., a mountain peak or tree) where we cannot access the base, and therefore cannot measure the full horizontal distance that separates us from the object we wish to measure. The general premise of the remote method is illustrated in figure \(9.9 .\) If we can use a clinometer (a device for determining the angle between horizontal and a sight-line to an object of interest) to determine the sight- line angles to the object of interest \(p\) from two different places \(b 1\) and \(b 2,\) and we know the distance between those places \(l,\) we can use trigonometry and algebra to determine the desired height \(H\). Devise a strategy for measuring \(H\) from the information gathered at \(b 1\) and \(b 2 .\) Derive and justify a formula that can be used for this task.

Short answer

The height of the object is given by \(H = \frac{l}{2}\sqrt{\tan(\alpha)\tan(\beta)}\) where \(\alpha\) and \(\beta\) are the sight-line angles from two different places that are a distance \(l\) apart, and \(l\tan(\alpha)\tan(\beta)\) is not zero.

Step-by-step solution

Understand the Trigonometric Principles

Firstly, try to visualize the problem as two right triangles with a common height. The trigonometric relationships within a right triangle are the primary tool needed to solve this problem. The sightline angles, or the angles formed by the line to the object from the ground, are the angles of interest in these triangles.

Set up the Equations from Each Position

From the two different locations, we form two equations using the tangent of the angles of sight. The tangent of an angle in a right triangle is the ratio of the side opposite the angle to the side adjacent. From position \(b_1\), with the sight-line angle being \(\alpha\), the height of the object \(H\) is related to the distance \(x\) (from \(b_1\) to the base of the object) by \(\tan(\alpha) = \frac{H}{x}\). From position \(b_2\), with the sight-line angle being \(\beta\), and the distance from \(b_2\) to the base of the object being \((l-x)\), the relation is \(\tan(\beta) = \frac{H}{l-x}\).

Solve for the Height

These two equations can be solved simultaneously to isolate \(H\). By multiplying these two equations together, we get \(H^2 = x(l-x)\tan(\alpha)\tan(\beta)\). If \(l\tan(\alpha)\tan(\beta)\) is not zero, we can rearrange the equation to get \(H = \sqrt{x(l-x)\tan(\alpha)\tan(\beta)}\). Since we do not know \(x\), we set its value as \(\frac{l}{2}\) to maximize the value of \(H\) (as it makes \(x(l-x)\) the largest). Therefore, the height is \(H = \frac{l}{2}\sqrt{\tan(\alpha)\tan(\beta)}\).

Key concepts

Right Triangle Trigonometry

Right triangle trigonometry is a fundamental tool for measuring distances, especially when accessing direct measurement isn’t feasible. A right triangle has one angle equal to 90 degrees. The trigonometric ratios like sine, cosine, and tangent are derived from these types of triangles.
In the context of our distance measurement problem, we primarily use the tangent function. The tangent of an angle is the ratio of the opposite side to the adjacent side in a right triangle.

• For example, if you know the angle of elevation from a certain point and you want to find out the height of an object, you would set up a right triangle and use the tangent of the angle.
• Formula: If you have a right triangle where the angle is α, the opposite side is the height H, and the adjacent side is the distance x from the base, then the tangent is expressed as: \[ \tan(\alpha) = \frac{H}{x} \]
• Application: Knowing this can help calculate the height of an object if you can measure the angle and the distance.

Understanding these relationships enables us to create equations that relate angles to distances, which are crucial in solving real-world measurement problems using trigonometry.

Clinometer Usage

A clinometer is a handy device widely used in trigonometry to measure angles of elevation or depression from a horizontal plane. It is essential for estimating heights indirectly.
To use a clinometer in a field scenario, you look through the device at the top of the object you wish to measure and read off the angle. This simple tool thus provides a way to acquire one of the critical components needed when measuring height indirectly.

• How it works: Adjust the clinometer and sight through it over a horizontal reference to the object's peak.
• Angle Measurement: Read the angle formed between your line of sight and the horizontal.
• Practical Use: Use these angles to compute the height of the object along with other known distances using trigonometric equations.

Using a clinometer simplifies the process and often complements the tools you have for practical mathematics and trigonometry problems.

Problem Solving in Mathematics

Problem-solving in mathematics, specifically with trigonometry and geometry, involves identifying relationships between known and unknown quantities and using mathematical principles to find solutions. This approach is invaluable in distance measurement problems where direct methods cannot be used.
When faced with such a problem:

• Identify Known Quantities: such as the distance between two observation points or angles obtained from instruments like a clinometer.
• Model the Scenario: Visualize the problem geometrically using diagrams. This helps to establish connections.
• Apply Mathematical Tools: Utilize algebra and trigonometric formulas to form equations based on your understanding of these relationships.
• Simultaneous Equations: Often, more than one equation is used to solve for an unknown. It might require solving these equations together.

These strategies are key when attempting to solve complex problems and ensure that mathematical theories provide viable and accurate solutions.

Algebraic Formulas in Geometry

Algebraic formulas are pivotal in solving geometric problems, especially in trigonometry, where they establish relationships between varying quantities. In distance measurement challenges, they are used to connect angles, distances, and heights.
In the scenario described, we use algebraic manipulation to derive a formula for height when given certain angles and a line of known distance.

• Formulate and Manipulate: Start by writing equations based on trigonometric principles. For example, \[ \tan(\alpha) = \frac{H}{x} \] and \[ \tan(\beta) = \frac{H}{l-x} \]
• Combine and Solve: Multiplying or dividing such equations helps to eliminate unknowns, making it easier to isolate the desired variable, usually H in these problems.
• Optimization: Changing other variables such as x in our problem to maximize or simplify the calculation of H demonstrates how algebra prepares us to handle diverse scenarios.

The precision and flexibility of algebraic formulas are what make them so powerful in geometry and mathematics as a whole.