# A figure consists of a semicircle with a rectangle on its diameter . Given the perimeter of the figure find its dimensions in order that the area may be maximum.

let the radius of the circle be r and the height of the rectangular part formed on the diameter of circle be h.
let the area of the figure be

since the perimeter of the figure is fixed , let it be 2k.

substituting in eq(1):

now to find the critical points let us differentiate eq(3) wrt r:

now to check  the maxima or minima we will again differentiate the equation (4):
$\frac{{d}^{2}A}{d{r}^{2}}=\frac{d}{dr}\left(\frac{dA}{dr}\right)\phantom{\rule{0ex}{0ex}}=\frac{d}{dr}\left(-\pi r+2k-4r\right)\phantom{\rule{0ex}{0ex}}=-\pi -4\phantom{\rule{0ex}{0ex}}⇒\frac{{d}^{2}A}{d{r}^{2}}<0$
thus $r=\frac{2k}{\pi +4}$ must be a relative maximum.
since there is only one critical point , a relative maximum is also an absolute max.
thus $r=\frac{2k}{\pi +4}$ gives the maximum area in the problem.
the corresponding value of the height is
$h=k-\frac{\pi r}{2}-r\phantom{\rule{0ex}{0ex}}=k-\frac{\pi }{2}.\frac{2k}{\pi +4}-\frac{2k}{\pi +4}\phantom{\rule{0ex}{0ex}}=k-\frac{\pi k}{\pi +4}-\frac{2k}{\pi +4}\phantom{\rule{0ex}{0ex}}=\frac{k\pi +4k-\pi k-2k}{\pi +4}\phantom{\rule{0ex}{0ex}}=\frac{2k}{\pi +4}$

hope this helps you

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