A rectangle is inscribed with its base on the x-axis and its upper corners of the parabola y= 9-x^2. What are the dimensions of such a rectangle with the greatest possible area?
Width? =
Height? =
The hardest part of optimization problems is set up. So this rectangle's area is W*H like you said. If you can't imagine the parabola well then perhaps draw it out on paper. It's an inverted parabola shifted up 9 units. It's symmetric to the line x=0, which is nice.
The height is clearly going to be dependent on y, because the top corners touch the parabola. So what about the width? Well you need a way of expressing this length in terms of an x value. It's not simply x, because that doesn't cover the whole length of the base. Can you see the way to write the length of the whole base for a given that the bottom right corner is at (x,0)?