Problem 63 - Calculus - AP Edition

Question

The side of a cube is measured with a possible percentage error of \(\pm 2 \% .\) Use differentials to estimate the percentage error in the volume.

Step-by-Step Solution

Answer

The estimated percentage error in the volume is \(\pm 6\%\).

Step 1: Understand the Problem

We need to find the percentage error in the volume of a cube when the side length is measured with a possible percentage error of \(\pm 2\%\). We will use the concept of differentials to estimate this.

Step 2: Recall the Formula for Volume

The volume \(V\) of a cube with side length \(s\) is given by the formula: \[ V = s^3 \]

Step 3: Differentiate the Volume Formula

We find the differential of the volume \(V\) with respect to the side \(s\), which is: \[ dV = 3s^2 \, ds \] This equation shows how changes in the side length \(s\) affect changes in the volume \(V\).

Step 4: Relate Differential to Percentage Error

To find the relative change in volume \(\frac{dV}{V}\), substitute the expressions for \(V\) and \(dV\): \[ \frac{dV}{V} = \frac{3s^2 \, ds}{s^3} = \frac{3 \, ds}{s} \] This simplifies to \(3 \, \frac{ds}{s}\), which represents the relative change in volume.

Step 5: Apply the Given Percentage Error in Side Length

The percentage error in the side length \(s\) is \(\pm 2\%\), or \( \frac{ds}{s} = \pm 0.02 \). Substitute this into the relative change in volume: \[ \frac{dV}{V} = 3 \, (\pm 0.02) = \pm 0.06 \]

Step 6: Convert Relative Change to Percentage Error

The relative change of \(\pm 0.06\) translates to a percentage error in volume of \(\pm 6\%\). Therefore, the estimated percentage error in the volume is \(\pm 6\%\).

Key Concepts

Percentage Error

Percentage error is a way to understand how much a measured or calculated value deviates from the exact or standard value. It's a crucial concept in fields like engineering, physics, and everyday tasks involving measurement. The idea is to express the error as a percentage of the exact value, providing an easy-to-read accuracy measure. For example, if a side length of a cube is measured to have a percentage error of \(\pm 2\%\), it means that the actual value may be more or less than the measured value by 2%. To find the percentage error, use the formula: \[ \text{Percentage Error} = \left(\frac{\text{Error}}{\text{True Value}}\right) \times 100\% \] In situations like the cube, once we find how much an error in measurement affects the final result (such as volume), we can better understand the reliability of our measurements. Possible sources of percentage errors include poor measurement techniques, instrument limitations, or environmental factors.

Volume of a Cube

The volume of a cube is a measure of the amount of space enclosed within its sides. For any cube, knowing the length of one side allows you to calculate the volume using the formula \(V = s^3\), where \(s\) represents the side length. Cubes are unique because all their sides are equal in length, making calculations straightforward. Understanding cube volume is fundamental not only in mathematics but also in real-world applications like packaging, architecture, and more. For example, if you know a cube’s side measures 3 cm, its volume is \(3^3 = 27 \text{ cm}^3\). Calculating volume with precision is important, especially when dealing with bulk materials or spaces, where even small errors can result in significant discrepancies.

Error Estimation

Error estimation, particularly in calculus, involves determining how errors in measurements can affect computations. Using differentials is a common method for estimating errors because it involves analyzing how small changes in one quantity affect another. In calculus, differentials help approximate how changes in variables lead to changes in functions. Suppose we measure the side of a cube with an error margin; this error will affect the cube's volume. The differential \(dV\) is used to analyze this. It's derived from the formula \(dV = 3s^2 \, ds\), indicating how a change in the side \(s\) affects the volume \(V\). For accurate error estimations, it's essential to consider the sensitivity of your calculations. A simple way is to relate differential changes to percentage errors, giving a clearer idea of potential inaccuracies. By knowing how errors propagate, decisions can be made more confidently, whether in engineering designs, scientific models, or daily measurements.