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Chapter 4: Problem 44

The area of an equilateral triangle with side 20\(\pm 0.5 \mathrm{cm} .\)

Short Answer

Expert verified
The area of the equilateral triangle is computed as \(A = \frac{\sqrt{3}}{4} \cdot s^2\). Using the side length of 20 cm, the area is \(\frac{\sqrt{3}}{4} \cdot (20)^2\). Because there is an uncertainty of 卤0.5 cm in the side measurement, the area should also be presented with its uncertainty. Therefore, the area of the equilateral triangle is between the computed minimum area \(\frac{\sqrt{3}}{4} \cdot (19.5)^2\) and maximum area \(\frac{\sqrt{3}}{4} \cdot (20.5)^2\).

Step by step solution

01

Compute Area With Given Side Length

Use the given length of the side (20 cm) in the formula \(A = \frac{\sqrt{3}}{4} \cdot s^2\). So the area \(A = \frac{\sqrt{3}}{4} \cdot (20)^2 \).
02

Compute the Minimum and Maximum Area

Compute the area for the minimum side length (20 - 0.5 = 19.5 cm) and the maximum side length (20 + 0.5 = 20.5 cm) using the same formula. So the minimum area \(A_min = \frac{\sqrt{3}}{4} \cdot (19.5)^2\), and the maximum area \(A_max = \frac{\sqrt{3}}{4} \cdot (20.5)^2\).
03

Express the Area with its Uncertainty

The area of the triangle should be represented with its uncertainty, which is the range between the minimum and maximum areas obtained in Step 2. It can be expressed as \(A_{avg} \pm Uncertainty\) where \(A_{avg}\) is the average of \(A_min\) and \(A_max\), and Uncertainty is half of the range between \(A_min\) and \(A_max\).

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Key Concepts

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

Equilateral Triangle
An equilateral triangle is a unique type of triangle where all three sides are of equal length. This geometric shape is highly symmetrical and has the property that all of its internal angles are equal, measuring 60 degrees each. In understanding such a triangle, this equal-sided and equal-angle characteristic helps in simplifying many computations related to its properties.
  • Each side is equal: If one side is known, the others are equally the same.
  • All angles are 60 degrees, which makes using trigonometric calculations more straightforward.
This simplicity is particularly helpful when you need to calculate geometric properties, like the area. With knowledge of just one side, you can uncover almost any aspect of the triangle. This is an important point as we move forward to the next concept, area calculation.
Area Calculation
Calculating the area of an equilateral triangle can be done with the formula \(A = \frac{\sqrt{3}}{4} \cdot s^2\), where \(s\) is the side length of the triangle. This formula is derived from understanding the properties of equilateral triangles and the ratios involved.
  • "\(\sqrt{3}/4\)": This is a constant that comes from understanding the geometry of a triangle and its height relative to its base.
  • "\(s^2\)": Squaring the side length captures the way area scales with the square of the sides.
To effectively use this formula in real-world applications, one must accurately measure the side length. Even a small deviation can significantly affect the resulting area. That leads us directly to our next topic, uncertainty in measurements.
Uncertainty in Measurements
In practical scenarios, measurements can never be perfectly accurate. There is always some level of uncertainty, which affects our calculations. In the context of an equilateral triangle's area, measurement uncertainty can change the computed area significantly.
  • "Uncertainty range": This is the range within which the true measurement likely exists. For example, 20 cm \(\pm\) 0.5 cm means the side could be as small as 19.5 cm or as large as 20.5 cm.
  • "Calculating uncertainty": By finding the minimum and maximum possible areas with the extreme side lengths, you establish a range of potential areas.
  • "Expressing with uncertainty": Represent the area as \(A_{avg} \pm Uncertainty\). \(A_{avg}\) is the average of the computed minimum and maximum areas, while "uncertainty" itself is half the difference between them.
Understanding and factoring in uncertainty ensures that your calculations are both accurate and realistic, particularly when precise measurements are critical. This practice is essential in scientific and engineering domains where reliable data is crucial.

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