A rectangle is inscribed in a semicircle of radius $R$ with base on th…

Question

A rectangle is inscribed in a semicircle of radius $R$ with base on the diameter. The **maximum area** is:

Accepted 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 = $2 \cdot (R/\sqrt 2)(R/\sqrt 2) = R^2$.

Answer

$2R^2$

Answer

$R^2/2$

Answer

$\pi R^2/2$

A rectangle is inscribed in a semicircle of radius $R$ with base on the diameter. The maximum area is:

Keep practising Unit 4-5: Applications of Derivatives & Curve Sketching