A rectangle is inscribed in a semicircle of radius 6 cm, with one side of the rectangle along the diameter. The maximum possible area of the rectangle is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 36 cm2
Let the centre of the diameter be the origin. A rectangle with base from −x to x on the diameter has its top corners on the semicircle, so its height is \(\sqrt{36 - x^{2}}\). The area is
\( A = 2x\sqrt{36 - x^{2}},\ 0 \lt x \lt 6 \)
It is easier to maximise \(S = A^{2}\):
\( S = 4x^{2}(36 - x^{2}) = 144x^{2} - 4x^{4} \)
\( S' = 288x - 16x^{3} = 16x(18 - x^{2}) \)
So \(x^{2} = 18\), \(x = 3\sqrt{2}\). Also S′ > 0 before and S′ < 0 after this point, so it gives the maximum. Then the height is \(\sqrt{36 - 18} = 3\sqrt{2}\), and
\( A = 2 \times 3\sqrt{2} \times 3\sqrt{2} = 36 \) cm2
Why the other options are incorrect
Option (1) 72 cm2 — Incorrect. This is the largest rectangle in the full circle of radius 6 (a square of diagonal 12), twice the semicircle answer.
Option (2) 18 cm2 — Incorrect. This uses the half-base x = \(3\sqrt{2}\) times the height, forgetting that the base is 2x.
Option (3) 24 cm2 — Incorrect. The area reaches 36 cm2 at x = \(3\sqrt{2}\), so 24 is not the maximum.