Problem 22
Question
Surveying A surveyor standing 50 feet from the base of a large tree measures the angle of elevation to the top of the tree as \(71.5^{\circ} .\) How accurately must the angle be measured if the percent error in estimating the height of the tree is to be less than \(6 \% ?\)
Step-by-Step Solution
Verified Answer
The angle must be measured to an accuracy of approximately ±1.2° to ensure that the percent error in estimating the height of the tree is less than 6%.
1Step 1: Define the Problem
The surveyor is 50 feet from the base of the tree and the angle of elevation is given as \(71.5^{\circ}\). Considering the distance and the angle to be a right triangle, the height (h) of the tree can be defined using trigonometric tangent function as: \(h = 50 \cdot \tan(71.5^{\circ})\)
2Step 2: Define the Maximum Allowable Error
The problem states that the percent error in the estimated height of the tree should be less than 6%. Therefore, the maximum allowable change in height (Δh) can be calculated as follows: \(Δh = 0.06 * h\)
3Step 3: Compute the Height and Maximum Allowable Error
Calculate the height of the tree using the initial angle, and then use this to determine the maximum allowable change in height.
4Step 4: Determine the Required Accuracy of the Angle
Knowing the tree's height and the maximum allowable error can help us find the new angle of elevation that would yield a tree height within the error bounds. Find this new angle using the inverse tangent function (arctan or tan^(-1)), and derive the desired accuracy of the angle from the difference between this new angle and the initial one.
Key Concepts
Angle of ElevationSurveying TechniquesPercent Error
Angle of Elevation
The concept of the angle of elevation is an essential part of trigonometry, especially when it comes to real-world applications like surveying.
It is the angle formed between a horizontal line from the observer and the line of sight to an object above the horizontal line.
In simpler terms, when you look up at a tree or a building, the angle your line of sight makes with that object is the angle of elevation.Understanding the angle of elevation helps in creating right triangles that involve the length of the baseline and the height of the object being measured.
For example, if you know the distance from you to the base of a tree and the angle of elevation to the top of a tree, you can calculate the height of the tree using basic trigonometry.
Here, the tangent function is helpful because it relates the opposite side (height of the tree) to the adjacent side (distance to the tree).
Using the formula:
It is the angle formed between a horizontal line from the observer and the line of sight to an object above the horizontal line.
In simpler terms, when you look up at a tree or a building, the angle your line of sight makes with that object is the angle of elevation.Understanding the angle of elevation helps in creating right triangles that involve the length of the baseline and the height of the object being measured.
For example, if you know the distance from you to the base of a tree and the angle of elevation to the top of a tree, you can calculate the height of the tree using basic trigonometry.
Here, the tangent function is helpful because it relates the opposite side (height of the tree) to the adjacent side (distance to the tree).
Using the formula:
- \[h = d \cdot \tan(\theta)\]
- \(h\) represents the height of the tree,\(d\) is the distance to the tree, and \(\theta\) is the angle of elevation.
Surveying Techniques
Surveying techniques are crucial in accurately measuring distances, angles, and heights without the need for direct measurement.
These techniques often rely on trigonometric principles, particularly when determining the height of tall structures or natural elements like trees. One such technique, used by surveyors, involves positioning themselves at a known distance from the object whose height is to be measured and determining the angle of elevation to the top of the object.
Thanks to trigonometry, once this angle and the baseline distance are known, the height of the object can be calculated accurately.
This method is non-intrusive and efficient, making it invaluable in architecture, engineering, and even archeology. Modern advancements have introduced advanced tools like theodolites and laser rangefinders, ensuring even greater precision in measurements.
However, the fundamental principle remains the same: leveraging the relationships within right triangles through trigonometric functions to achieve accurate measurements.
In the case of our exercise, understanding the relation between the baseline distance, angle of elevation, and height of the tree is essential to apply this surveying technique effectively.
These techniques often rely on trigonometric principles, particularly when determining the height of tall structures or natural elements like trees. One such technique, used by surveyors, involves positioning themselves at a known distance from the object whose height is to be measured and determining the angle of elevation to the top of the object.
Thanks to trigonometry, once this angle and the baseline distance are known, the height of the object can be calculated accurately.
This method is non-intrusive and efficient, making it invaluable in architecture, engineering, and even archeology. Modern advancements have introduced advanced tools like theodolites and laser rangefinders, ensuring even greater precision in measurements.
However, the fundamental principle remains the same: leveraging the relationships within right triangles through trigonometric functions to achieve accurate measurements.
In the case of our exercise, understanding the relation between the baseline distance, angle of elevation, and height of the tree is essential to apply this surveying technique effectively.
Percent Error
Percent error is a measure of how accurate a measurement is, expressed as a percentage of the actual value.
It is crucial when exact precision is required, such as in scientific measurements or in surveying large structures.To calculate percent error, you compare the measurement error to the true value and then multiply by 100 to convert it into a percentage.
This helps in understanding the proportion of error relative to the size of the measurement, providing a sense of how significant the error is.
The percent error formula is:
The requirement that this error be less than 6% means the angle of elevation measurements must be very precise.
Even a small deviation in this angle would lead to an error above the allowable threshold—making the surveying results unreliable.Therefore, percentage error not only ensures the accuracy of the measurements but also dictates the precision required during the measurement process, thereby impacting the reliability of the entire surveying task.
It is crucial when exact precision is required, such as in scientific measurements or in surveying large structures.To calculate percent error, you compare the measurement error to the true value and then multiply by 100 to convert it into a percentage.
This helps in understanding the proportion of error relative to the size of the measurement, providing a sense of how significant the error is.
The percent error formula is:
- \[ \text{Percent Error} = \left( \frac{\text{measured value} - \text{true value}}{\text{true value}} \right) \times 100 \% \]
The requirement that this error be less than 6% means the angle of elevation measurements must be very precise.
Even a small deviation in this angle would lead to an error above the allowable threshold—making the surveying results unreliable.Therefore, percentage error not only ensures the accuracy of the measurements but also dictates the precision required during the measurement process, thereby impacting the reliability of the entire surveying task.
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