91Ó°ÊÓ

The angle of elevation is a crucial concept in trigonometry, especially when it comes to measuring heights indirectly. It is defined as the angle between the horizontal line from the observer and the line of sight to an object above the horizontal line. For example, when a surveyor looks up at the top of a building from a point on the ground, the angle they tilt their line of sight upwards is the angle of elevation.

In the given problem, the surveyor's angle of elevation to the top of the building is 75 degrees. This measurement is important because it allows the surveyor to calculate the height without directly measuring it. Changes or errors in this angle may significantly affect the estimate of the building's height, making its accurate measurement vital.

The tangent function plays an essential role in trigonometry, especially when dealing with right-angled triangles. In our scenario, the tangent of the angle of elevation helps calculate the height of the building. The tangent of an angle in a right triangle is defined as the ratio of the opposite side (in this case, the height of the building) to the adjacent side (the distance from the building's base).

Mathematically, if we denote the angle of elevation as \( \theta \), the formula using the tangent function is:

• \( h = \tan(\theta) \times \text{distance from base} \)

So, with the standing distance being 30 feet and \( \theta \) being 75 degrees, the height \( h \) comes from \( h = \tan(75^{\circ}) \times 30 \). This formula highlights the role of the tangent function in estimating the building's height.

The tangent function's sensitivity is why precise angle measurements are necessary. A small error in \( \theta \) could lead to considerable changes in the calculated height.

Percentage error quantifies the accuracy of a measurement by comparing the error in measurement to the true value, expressed as a percentage. It gives an indication of the size of the error relative to the measured value.

In the context of this problem, the percentage error is used to evaluate how accurate the height estimation of the building is based on the angle measurement. The goal is to ensure that any error in the angle measurement leads to less than a 4% error in the calculated height. This is crucial for precision in practical applications.

• Formula: \( \frac{\Delta h}{h} \times 100\% \)

where \( \Delta h \) is the error in height measurement and \( h \) is the estimated height. In order to embed accurate measurements into calculations, this step ensures the integrity of derived values.

Approximating small changes using derivatives is a powerful mathematical tool, particularly relevant when dealing with minute variations in values like angles. This technique assumes the function is approximately linear over small intervals, simplifying complex equations.

In this problem, we use the derivative of the tangent function to approximate the changes in height resulting from small changes in angle (\( \Delta \theta \)). For small \( \Delta \theta \), the change in tangent is approximated by its derivative:

• \( \Delta(\tan(\theta)) \approx (\sec^2(\theta)) \Delta\theta \)

Here, \( \sec^2(\theta) \) is the derivative of \( \tan(\theta) \). Using this approximation, we determine how accurate angles must be to keep the percentage error in height under 4%.

This derivative simplifies calculations and provides an easier path to understanding how small variations impact the overall results, especially in engineering and physics contexts.