Multiple choice

An engineer needs to design a rectangular display board that must fit entirely within a semi-circular frame of radius 5 cm, maximising the area of the board. What will be the ratio of the board's shorter side to its longer side when its area is the largest possible?

  1. 1 : 3

  2. 2 : 5

  3. 1 : 2

  4. 2 : 3

  5. 3 : 5
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let the rectangle have width 2x and height y. It is inscribed in a semicircle of radius R=5. By Pythagorean theorem, x^2 + y^2 = R^2 = 25. Area A = 2xy = 2x * sqrt(25 - x^2). To maximize, dA/dx = 0. A^2 = 4x^2(25 - x^2) = 100x^2 - 4x^4. Derivative = 200x - 16x^3 = 0 => x^2 = 200/16 = 12.5. Then y^2 = 25 - 12.5 = 12.5. So x = y. The sides are 2x and y = x. Ratio of shorter side (x) to longer side (2x) is 1:2.

AI explanation

The area of the rectangular board inscribed in a semicircle of radius 5 is maximized when the height to length ratio corresponds to the half-angle of the subtended arc. Through the derivative method applied to the area equation A = x times the square root of (25 minus x squared), the optimal width x is found to be 5 divided by the square root of 2. This establishes the longer side as 5 times the square root of 2, resulting in the shorter side to longer side ratio of 1 to 2.

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