A rectangle is inscribed in the semicircle y = sqrt[4 - x^2]. Find its largest possible area.
I think I am suppose to take the derivative, but what do I do next??
Center of the circle was (0,0)
We take a point (x,0) as one of the edge of rectangle
The edge just above (x,0) will have
y coordinate lying on the semicircle and given by
$\displaystyle \sqrt{4-x^2}$ , hence its coordinate is
$\displaystyle (x , \sqrt{4-x^2}) $
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Make sure you get the above part before you continue.
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Area of this rectangle will be given by
$\displaystyle = 2x \times (\sqrt{4-x^2})$
...............{2 times of x because the length will lie on both side of the axis}
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if you don't get the above thing try drawing the diagram.
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This area needs to be max. so try finding the maximum
value of Area(x)
$\displaystyle A(x) = 2x\sqrt{4-x^2} $
for maxima
$\displaystyle
\frac{d}{dx}A(x) = 0 $
From above get the value of x for which A(x) is 0 . try this
value on A(x) to get max. area.
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Be sincere enough to read below after you have solved.
Spoiler:
Now find area(use +ve x).