# Math Help - Calculus - maximizing area

1. ## Calculus - maximizing area

A rectangle is inscribed with its base on the x-axis and its upper corners on the parabola y = 10−x2. What are the dimensions of such a rectangle with the greatest possible area?

2. We first formulate everything in mathematical terms: if $x$ is the distance from the origin of one lower corner of the rectangle, then the base of the rectangle is $2x$, and the height of the rectangle is $10-x^2$. Its area is therefore

$A=2x\cdot(10-x^2).$

To maximize $A$, we remember that $A$ is differentiable everywhere and approaches $0$ at the boundary points of its domain, and therefore that $A$ will attain an extremum at some point at which $\frac{dA}{dx}=0$.