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Rectangle mirror
Rectangle mirror
Calculus Optimization Problem #4?


Final problem >_> man homework is hard.

A mirror in the shape of a rectangle capped by a semicircle is to have a perimeter of 22m. Find the rads of the semicircle so that the mirror has maximum area.

Thanks in advance again.

area s=s1+s2, where
s1=a*b is the area of rectangular part,
s2=0.5*pi*r^2 is area of round part;
a=2*r;
perimeter p = a +2b +pi*r =22, hence b=(p –2*r -pi*r)/2;
plugging b into s:
s = (2*r)*(p –2*r -pi*r)/2 +0.5*pi*r^2;
ds/dr = p –4*r -2pi*r +pi*r = p -(pi +4)*r =0, hence
r=p/(4+pi) = 22/(4+pi) =3.081 m; (s=33.886 m^2);

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