Random Squares

A factory manufactures wooden squares whose side length should be one meter. However, the side length of the squares is not always \(\SI{1}{\meter}\) but varies uniformly between \(\SI{1}{\meter}\) and \(\SI{1.1}{\meter}.\) Note that the height and the breadth of each square piece remain equal to each other.

Two inspectors, Alice and Bob, want to calculate the average area of a square.

  • Alice claims that the average area is \(\SI{1.1025}{\meter\squared}\) because the average side length is \(\SI{1.05}{\meter}\) and \(1.05 \times 1.05 = 1.1025.\)
  • Bob claims that the average area is \(\SI{1.1050}{\meter\squared}\) because the the area is between \((\SI{1}{\meter} \times \SI{1}{\meter})\) and \((\SI{1.1}{\meter} \times \SI{1.1}{\meter}),\) and \(\frac{1 \times 1 + 1.1 \times 1.1}{2} = 1.105\).

Who is correct?

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