Problem 62 - Calculus - AP Edition

Question

The side of a square is measured with a possible percentage error of \(\pm 1 \% .\) Use differentials to estimate the percentage error in the area.

Step-by-Step Solution

Answer

The estimated percentage error in the area of the square is \( 2\% \).

Step 1: Understand the Problem

We need to estimate the percentage error in the area of the square given a percentage error in the measurement of its side. We'll use differentials to perform this estimation.

Step 2: Identify the Formula for Area

The area of a square is calculated using the formula: \[ A = s^2 \] where \( s \) is the length of one side of the square.

Step 3: Calculate the Differential of the Area

To find the differential \( dA \), we differentiate the area \( A = s^2 \) with respect to \( s \). The derivative \( \frac{dA}{ds} = 2s \), thus: \[ dA = 2s \, ds \]

Step 4: Relate Differential to Percentage Error

The percentage error in the side length is \( \pm 1\% \), meaning \( ds = 0.01s \). Substituting \( ds \) into the differential equation, we get: \[ dA = 2s \times 0.01s = 0.02s^2 \]

Step 5: Calculate Percentage Error in Area

Estimate the percentage error in area, which is given by \( \frac{dA}{A} \times 100\% \). Given \( A = s^2 \), \[ dA = 0.02s^2 \], the percentage error in area becomes: \[ \frac{dA}{A} = \frac{0.02s^2}{s^2} = 0.02 \rightarrow 0.02 \times 100\% = 2\% \]

Key Concepts

Understanding Percentage Error

Percentage error is a way to express the accuracy of a measurement. It shows you how much the measured value deviates from the true value, expressed as a percentage. In the context of the exercise, we have calculated how a 1% error in measuring the side of the square affects the accuracy of the computed area of the square.
The percentage error can be computed using the formula:

• Percentage Error = \( \left( \frac{\text{Error in Measurement}}{\text{True Value}} \right) \times 100\% \)

In simple terms, though the side length was not measured perfectly, the allowed error in that measurement can give us insight into how much the computed area might vary from the actual area.
This concept is useful when you need to know the reliability of your measurements or calculations.

Calculating the Area of a Square

To calculate the area of a square, all you need is the side length of the square. The formula is:

• Area = \( s^2 \)

Here, \( s \) is the side length of the square.
The simplicity of the formula means that as long as you know the length of one side, you can compute the area by squaring that length. In our exercise, this step is crucial as we are using this formula to determine what happens to the area when the side length has an error.
Understanding this formula is key when computing how changes in side measurements affect the overall area.

The Role of Derivatives in Measuring Change

Derivatives allow us to understand how a change in one quantity affects another. When we're speaking about the area of a square with a known side length, the derivative helps us know what happens to the area when there's a slight change in that side length.
If we differentiate the area with respect to the side length, we get:

• \( \frac{dA}{ds} = 2s \)

This tells us the rate at which the area changes with respect to changes in the side length. With this derivative, you can calculate small changes in the area (\( dA \)) when there's a tiny change (\( ds \)) in the side length.
In this exercise, using derivatives helped us estimate how the allowed error in measuring the side translates to changes in the area.