Question

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

Short answer

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 ±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

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 \(A = \frac{\sqrt{3}}{4} \cdot (20)^2 \).

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 \(A_max = \frac{\sqrt{3}}{4} \cdot (20.5)^2\).

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 \(A_max\).