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Chapter 14: Problem 53

You plan to calculate the area of a long, thin rectangle from measurements of its length and width. Which dimension should you measure more carefully? Give reasons for your answer.

Short Answer

Expert verified
Measure the width more carefully, as it has a larger comparative impact on the area calculation in a long, thin rectangle.

Step by step solution

01

Understand the Area Formula

To calculate the area of a rectangle, we use the formula: \( \text{Area} = \text{Length} \times \text{Width} \). Understanding this formula is essential before determining which measurement should be more precise.
02

Identify Impact of Each Dimension

Consider the effect of changing one dimension slightly while keeping the other constant. For example, changing the length by a small amount changes the area by that small amount multiplied by the width. Similarly, changing the width results change by an amount multiplied by the length.
03

Evaluate the Influence of Each Dimension

Suppose the rectangle is long and thin. In such a case, the length is significantly larger than the width. Hence, any relative error in measuring the width could result in a comparatively larger percentage error in the calculated area, since the length magnifies this error during multiplication.
04

Choose the Dimension for Careful Measurement

Since the width is smaller and has a more significant relative impact on the area calculation due to the rectangle's shape, the width should be measured more carefully for a better accuracy of the area.

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Key Concepts

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

Rectangle Area Calculation
When calculating the area of a rectangle, grasping the formula is the initial step. The area is obtained by multiplying the rectangle's length by its width. This basic formula is not only straightforward but also crucial to understand fully when assessing the impact of measurement errors. For many practical purposes, having precise measurements is key because any error directly affects the area. Since the area depends on both dimensions being multiplied, even a small mistake at this stage can have a noticeable impact on your final result. By simply assembling the formula correctly, you lay a solid foundation for understanding any additive errors, especially in more complex shapes or applications.
Relative Error Impact
Understanding the concept of relative error is vital when dealing with areas of shapes like rectangles. Relative error assesses the size of the error in measurement compared to the true value.
  • A small relative error in a larger measurement can contribute significantly to the total area.
  • In our case of a long, thin rectangle, the width is usually the smaller dimension compared to the length.
When measuring, a slight inaccuracy in the smaller dimension (width) becomes a larger percentage error since the length remains constant and serves to magnify any errors in calculation. Therefore, in shapes where one dimension is dramatically larger than the other, prioritizing precision in measuring the smaller dimension can lead to much better overall accuracy in area calculations.
Geometry of Rectangles
In geometry, understanding a rectangle's properties is essential to accurately analyzing measurement strategies. A rectangle is a four-sided polygon with opposite sides equal, giving it a straightforward shape but one that can present practical challenges during measurement.
  • The longer side often contributes more directly to changes in the area, especially if it is significantly greater than the other dimension as in long thin rectangles.
  • The smaller side, or width, can introduce larger relative errors in calculations since any mismeasurement represents a more substantial portion of this dimension.
In the configuration of long, narrow rectangles, minimizing error in the width can be crucial. This happens because any discrepancy will have a multiplying effect when both dimensions are used to calculate the area, emphasizing the need to carefully and precisely measure smaller dimensions.

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