Find the dimensions of the largest rectangle with sides parallel to the axes that can be inscribed in the x^2+4y^2=4 as shown in below.

Question

misty1212 · Answer

HI!!

misty1212 · Answer

guess this is a nice ellipse right?

anonymous · Answer

|dw:1428285653368:dw|

tkhunny · Answer

The AREA of such a rectangle would be $4\cdot x_{0}\cdot y_{0}$, just working in the first quadrant.

misty1212 · Answer

|dw:1428285756795:dw|

misty1212 · Answer

a point in the first quadrant will look like 
$$(x,\frac{\sqrt{4-x^2}}{2})$$

misty1212 · Answer

the area of that part will be 
$$A(x)=\frac{x\sqrt{4-x^2}}{2}$$ maximize that one

dan815 · Answer

|dw:1428285959909:dw|

dan815 · Answer

:)

dan815 · Answer

here is a trick!

dan815 · Answer

u know a square would be the biggest area inscribed in a circle so if we were to rescale the y axis or xaxis

misty1212 · Answer

@dan815  it is "inscribed" right?

dan815 · Answer

yes im just using the guidelines to draw a nice ellipse