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A rectangle has a base on the x-axis and its upper two vertices on the parabola.

y=10-x^2

what is the largest area the rectangle can have, and what are the dimensions of the rectangle with the largest area.

Set the base at (-a,a).

The height would then be 10 - a^2.

The area would then be 2a(10 - a^2).

That is, A(a) = 20a - 2a^3.

Taking the derivative gives 20 - 6a^2.

Setting this to 0 gives 20 = 6a^2, so 10/3 = a^2, so a = root(10/3).

That area is than 2*root(10/3)(10 - 10/3) = 2*root(10/3)(20/3) = (40/3)*root(10/3).

Rationalizing the denominator gives (40/3)*(root(30)/3) = root(30)*(40/9).

This would be roughly 5.5*4.4 = 24.2; found exactly gives 24.34322478.

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