Multiple choice

The sides of a rectangle are chosen at random each less than 10 cm, all such lengths being equally likely. The chance that the diagonal of the rectangle is less than 10 cm is

  1. $\dfrac{1}{10}$
  2. ${1}{20}$
  3. $\displaystyle \dfrac{\pi}{4}$
  4. $\displaystyle \dfrac{\pi}{8}$
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C Correct answer
Explanation

Sides x, y are in [0, 10]. Diagonal d = sqrt(x^2 + y^2) < 10 implies x^2 + y^2 < 100. This is the area of a quarter circle of radius 10 inside a 10x10 square. Probability = (pi * 10^2 / 4) / 10^2 = pi/4.

AI explanation

Let the sides of the rectangle be x and y, uniformly distributed between 0 and 10, making the area of the sample space 10 times 10, or 100. For the diagonal to be less than 10, the condition x squared plus y squared is less than 100 must hold, defining a quarter circle of radius 10 with an area of 25 times pi. The required probability is the ratio of the favorable area to the total area, which is 25 times pi divided by 100. Therefore, the result is pi divided by 4.

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