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Chapter 3: Problem 54

About how accurately should you measure the side of a square to be sure of calculating the area within 2\(\%\) of its true value?

Short Answer

Expert verified
Measure the side accurately within 1% of the true length.

Step by step solution

01

Understand the Problem

To find out how accurately the side of a square must be measured to ensure that the calculated area is within 2% of the true area, we need to relate the side length of the square to its area. The area of a square is given by the formula \( A = s^2 \), where \( s \) is the side length. We need to find the maximum permissible error in the side length such that the error in area remains within 2%.
02

Define the Formal Relationship

The true area is \( A_{true} = s^2 \). If the side length is measured incorrectly by a small amount \( \Delta s \), the corresponding measured area will be \( (s + \Delta s)^2 = s^2 + 2s \Delta s + (\Delta s)^2 \). The error introduced in the area \( \Delta A \) is \( 2s \Delta s + (\Delta s)^2 \). Since \( \Delta s^2 \) is very small compared to the other terms, we can often ignore it for small errors.
03

Set Up the Error Bound Equation

For error in area to remain within 2%, we need \( \frac{\Delta A}{s^2} \leq 0.02 \). Plugging in \( \Delta A \approx 2s \Delta s \) into the equation gives \( \frac{2s \Delta s}{s^2} \leq 0.02 \). This simplifies to \( \frac{2\Delta s}{s} \leq 0.02 \).
04

Solve the Inequality for Measurement Accuracy

Now solve for \( \Delta s \) by isolating it on one side. Multiply both sides of \( \frac{2\Delta s}{s} \leq 0.02 \) by \( s \) to get \( 2\Delta s \leq 0.02s \). Further simplify to \( \Delta s \leq 0.01s \). This means \( \Delta s \), the error in measurement of the side length, must not exceed \( 1\% \) of the true side length \( s \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Geometry
Geometry is a branch of mathematics that deals with shapes, sizes, and properties of space. When we discuss a square, we refer to a simple, four-sided shape where all sides are equal in length, and all interior angles are right angles.
To calculate the area of a square, we use the formula:
  • Area = side length 脳 side length or simply, \( A = s^2 \)
This formula helps us understand how the side length directly affects the area of the square. If the side length changes, the area changes significantly because the side length is squared. That's why measuring the side accurately is crucial when calculating the area.
Percentage Error
Percentage error is a way to express the accuracy of a measurement by comparing the error of a measured value to the true value. It is particularly helpful when you want to communicate how far a measured value deviates from the true or accepted value.
  • The formula to calculate percentage error is: \( \text{Percentage Error} = \left( \frac{\text{Error}}{\text{True Value}} \right) \times 100 \% \)
In our exercise, we need the percentage error in the area of a square to be within 2%. This means that the calculated area must be very close to the true area, within just a small fraction of it. Therefore, controlling the error in the side measurement is key to maintaining such a low percentage error in the area.
Measurement Accuracy
Measurement accuracy refers to how close a measured value is to the actual or true value. In the context of measuring the side of a square, it means how precisely we can determine the length of the side.
To ensure that the calculated area remains within 2% of the true area, as per the exercise, the measurement accuracy in the side length must be high. Specifically, the error in the side length (\( \Delta s \)) should not exceed 1% of the actual side length (\( s \)).
  • Improving measurement accuracy involves using better measuring tools or techniques to reduce external factors that may cause errors.
  • Consistency in the units of measurement also plays a crucial role in maintaining accuracy.
Area Calculation
Calculating the area of a given geometric shape involves understanding and applying specific formulas to convert the measurements into a value that represents the space within the shape. For a square, the area is calculated as the product of the side length with itself: \( A = s^2 \).
In practical terms, calculating the area accurately requires precise measurements of the side length. Inaccurate measurements can cause substantial discrepancies in the calculated area because the error is magnified when squared.
To keep the error within 2%, the side measurements must adhere to strict accuracy constraints. These constraints ensure that the error introduced during measurement does not exceed a permissible margin, which is crucial in real-life applications like construction, design, and land measurement.

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Most popular questions from this chapter

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