Problem 26 - Calculus of a Single Variable

Question

Area The measurement of the radius of a circle is 16 inches, with a possible error of \(\frac{1}{4}\) inch. (a) Use differentials to approximate the possible propagated error in computing the area of the circle. (b) Approximate the percent error in computing the area of the circle.

Step-by-Step Solution

Answer

The approximate propagated error in the area is \(16\pi\) square inches and the approximate percent error is approximately \(3.125%\)

Step 1: Find the Differential of the Area Function

The area function of a circle is \(A = \pi r^2\). Differentiating both sides with respect to \(r\), we get \(dA = 2\pi r dr\). This is the differential of \(A\), which represents the rate at which \(A\) changes for small changes in \(r\).

Step 2: Compute the Approximate Error in the Area

From the differential equation, \(dA = 2\pi r dr\), we substitute the values \(r = 16\) inches and the error rate \(dr = \frac{1}{4}\) inches to find \(dA\). Thus, \(dA = 2\pi (16) (\frac{1}{4})\).

Step 3: Calculate the approximate Percent Error

The percent error is given by \(\frac{Propagated Error}{Exact Value} × 100%\). The exact value is the precise area \(A = \pi r^2\) with \(r = 16\), and the propagated error is \(dA\). Calculate the percent error using these values.

Key Concepts

Differentials in Calculus

Differentials are a fundamental aspect of calculus, often used to approximate small changes in a function's output due to small changes in its input. Specifically, in the context of the area of a circle, if we denote the area by the function A(r) = \(\pi r^2\), where r is the radius, the differential dA can be understood as the change in the area for a slight change in the radius, represented by dr. By differentiating A with respect to r, we obtain dA = 2\pi r dr. This formula gives us a linear approximation of how the area changes as the radius varies, which is particularly useful when dealing with small changes.

Area of a Circle

The area of a circle, denoted by A, is a measurable attribute expressing the size of the surface enclosed by the circle. It is calculated using the formula A = \(\pi r^2\), with \pi being a mathematical constant approximately equal to 3.14159, and r the radius of the circle. Understanding this formula is crucial when assessing the impact of errors in the radius measurement on the calculated area, since the area is a function of the square of the radius, causing errors in the radius to affect the area to a greater extent.

Percent Error

Percent error is a measure of how accurate a calculation is, represented as a percentage. It's an expression of the magnitude of the difference between the measured or estimated value and the actual value, in relation to the actual value itself. The formula for percent error is \text{Percent Error} = \(\frac{\text{Propagated Error}}{\text{Exact Value}} \times 100\%\). It is particularly useful to assess the significance of an error in practical terms, as it provides a relative measure of accuracy rather than an absolute one. In exercises related to geometric measurements, like the area of a circle, calculating the percent error helps quantify the reliability of the computed area given imprecise radius measurements.

Measurement of Uncertainty

When measuring physical quantities, an inevitable element of uncertainty always arises. This measurement uncertainty can originate from various factors, including the precision of the measuring instrument or the method of measurement itself. Understanding the concept of uncertainty is crucial for interpreting scientific data accurately. Recognizing that the given radius of a circle, in this case, 16 inches with an error of \(\frac{1}{4}\) inch, contains uncertainty, allows students to calculate the propagated error and percent error, which lend insight into how reliable a calculated measurement, such as the area of a circle, truly is.