Problem 36 - Calculus of a Single Variable

Question

Approximating Function Values In Exercises \(37-40,\) use differentials to approximate the value of the expression. Compare your answer with that of a calculator.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

Answer

The degree of accuracy needed in the measurement of the angle, to ensure that the error is less than 6%, is approximately 0.00033 radians, or 0.019 degrees, when compared to a calculator result.

Step 1: Identify the Function

The tree and its distance are akin to a right-angled triangle. The tangent of angle \( \theta \) would be \( \frac{height}{base} \), where the base is the distance from the tree (50 feet) and height is the tree's height. Therefore, the height of the tree, as a function, is represented as \( h = 50 \cdot \tan(\theta) \).

Step 2: Calculate the Differential

Before finding the differential (dh), the implicit differentiation with respect to \( \theta \) must be found, which gives: \( \frac{dh}{d\theta} = 50 \cdot \sec^{2}(\theta) \). Multiplying by d\( \theta \), we get \( dh = 50 \cdot \sec^{2}(\theta) \cdot d\theta \).

Step 3: Approximate the Error

The problem states an error in height estimation of less than 6%, which is \(0.06h\). Set this equal to the value for 'dh' and rearrange for d\(\theta\). So, \(50 \cdot \sec^{2}(\theta) \cdot d\theta = 0.06h \). Substituting \( h = 50 \cdot \tan(\theta) \), we find \( d\theta = \frac{0.06 \cdot 50 \cdot \tan(\theta)}{50 \cdot \sec^{2}(\theta)} = 0.06 \cdot \tan(\theta)\cdot \cos^2(\theta) \). Plugging in \( \theta = 71.5^{\circ} \), one can find d\( \theta \).

Step 4: Comparison to Calculator Result

Substitute \( \theta = 71.5^{\circ} \) into d\( \theta \) to find the relative accuracy required in the measurement of the angle. Convert the result to degrees for comparison.

Key Concepts

Approximating Function Values

The concept of approximating function values is an essential technique in calculus and essential for situations where direct measurement or computation is difficult or impossible. This approach is based on using the concept of differentials, which are small changes in a function's output resulting from small changes in its input.

When you have a function that's not easily solvable using standard methods, differentials allow you to estimate the value of this function at a nearby point if you know the value at a specific point and the derivative at that point. In the context of surveying, as in the exercise, a surveyor can use differentials to estimate the height of a tree by understanding the small change in height that would result from a small change in the angle of elevation. This estimation helps bridge the gap between difficult-to-measure real-world values and their mathematical predictions.

Angle of Elevation

In practical terms, the angle of elevation is the angle between the horizontal plane and the line of sight from an observer to some point of interest above them, such as the top of a tree. This concept is crucial not only in surveying but also in navigation, architecture, and various fields of engineering.

Calculating the angle of elevation is fundamental in determining the height of objects without having to measure them directly. For instance, using trigonometric functions like tangent, which relates the angle of elevation to the ratio of the height of the object to the distance from the object, surveyors can calculate the height easily and efficiently. The angle of elevation is thus a pivotal reference for trigonometric calculations and can lead to approximations of height with a reasonable degree of precision when measured accurately.

Percent Error in Estimation

The percent error in estimation expresses the accuracy of an approximation as a percentage of the exact value. It's a vital concept in science, engineering, and any field that relies on measurements and calculations. The lower the percent error, the more accurate the approximation.

In the context of the given problem, the surveyor needs to determine the maximum allowable error in measuring the angle of elevation to guarantee that the percent error in the estimated height of the tree is less than 6%. This approach implies an understanding of the acceptable limits of error for the task at hand, which is essential for making sure the results of a calculation are within a useful range of accuracy. In many fields, there are standard acceptable percent error values that are used to determine the precision of instruments and the validity of results.

Implicit Differentiation

Implicit differentiation is a technique in calculus used when dealing with equations that define functions implicitly rather than explicitly. Explicit functions follow the form of 'y = f(x)', but for implicit functions, both variables are mingled together in an equation, without a clear subject of the formula.

This technique involves differentiating both sides of an equation with respect to one variable and solving for the derivative of the other. It is useful in cases where it is difficult or impossible to solve for one variable in terms of the other variable before differentiating. In our exercise, implicit differentiation helps us find the rate at which the height of the tree changes with respect to the angle of elevation, which is then used to approximate the error committed when measuring the angle within certain tolerances.