Multiple choice

$I:$ The sides of a rectangle are $(6-x)\ cm$ and $(x-3)\ cm$. If its area is maximum, then $x =4.5$ $II:$ The maximum area of the rectangle inscribed in a circle of radius $5\ cm$ is $50\ sq.cm$. Which of the above statements are true?

  1. only $I$
  2. only $II$
  3. both $I$ and $II$
  4. neither $I$ nor $II$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

I: Area = (6-x)(x-3) = -x^2 + 9x - 18. Max area at x = -b/2a = -9/-2 = 4.5. II: For a rectangle inscribed in a circle of radius R, max area is a square with side R*sqrt(2). Area = (5*sqrt(2))^2 = 50. Both are true.

AI explanation

For statement I, the area of the rectangle is given by A = (6-x)(x-3) = -x^2 + 9x - 18. To find the maximum area, we take the derivative and set it to zero, giving -2x + 9 = 0, which means x = 4.5; this statement is true. For statement II, the maximum area of a rectangle inscribed in a circle occurs when it is a square, so its diagonal equals the diameter of 10 cm. Using the Pythagorean theorem, s^2 + s^2 = 10^2, which gives s^2 = 50 sq. cm; this statement is also true. Therefore, both I and II are true.

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