alleryA rectangle is inscribed in a semicircle of radius $R$ with base on the diameter. The **maximum area** is:
A$R^2$ (when the rectangle is a square inscribed)
B$2R^2$
C$R^2/2$
D$\pi R^2/2$
Answer & Solution
Correct answer: A. $R^2$ (when the rectangle is a square inscribed)
Half-width = $x$, height $y = \sqrt{R^2 - x^2}$.
Area $A = 2xy = 2x\sqrt{R^2 - x^2}$.
We have $dA/dx = 2\sqrt{R^2 - x^2} - 2x^2/\sqrt{R^2 - x^2}$.
So $dA/dx = 0$.
⇒ $R^2 = 2x^2$ ⇒ $x = R/\sqrt 2$.
Then $y = R/\sqrt 2$, area = $2 \cdot (R/\sqrt 2)(R/\sqrt 2) = R^2$.
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