How to find the dimensions of a rectangle if its area is to be a maximum?

Solution 1:

The equation for the area of the rectangle (depending on the $x$-coordinate of the vertex on the right side of the $y$-axis) will be $A(x)=2x(18-x^2)$. Do you know how to determine which $x$ maximizes this? Note that we will only be considering $0<x<\sqrt{18}$.

Edit: The first thing we probably want to do for a problem like this is draw a picture. A general rectangle as described will look something like this:

Note that the height of the rectangle is $y$, and the width is $x-(-x)=2x$, so the area is $2xy$. We also know $y=18-x^2$, since $(x,y)$ lies on that parabola, so that gives us the expression I have above for area, and it's entirely in terms of $x$. We can also rewrite it as $A(x)=36x-2x^3$. Taking the derivative, we have $A'(x)=36-6x^2$. We need to find the positive solution to $A'(x)=0$, which will give us the maximum area. If $x_0$ is that value of $x$, then the dimensions will then be $2x_0$ by $18-x_0^2$.