Random Squares

A factory manufactures wooden squares whose side length should be one meter. However, the side length of the squares is not always 1 m\SI{1}{\meter} but varies uniformly between 1 m\SI{1}{\meter} and 1.1 m.\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 1.1025 m2\SI{1.1025}{\meter\squared} because the average side length is 1.05 m\SI{1.05}{\meter} and 1.05×1.05=1.1025.1.05 \times 1.05 = 1.1025.
  • Bob claims that the average area is 1.1050 m2\SI{1.1050}{\meter\squared} because the the area is between (1 m×1 m)(\SI{1}{\meter} \times \SI{1}{\meter}) and (1.1 m×1.1 m),(\SI{1.1}{\meter} \times \SI{1.1}{\meter}), and 1×1+1.1×1.12=1.105\frac{1 \times 1 + 1.1 \times 1.1}{2} = 1.105.

Who is correct?


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