# Inscribed Rectangle under a Parabola

- Author:
- cconrad

What is the area of the largest rectangle that you can inscribe between the parabola and the x-axis?

Using Geogebra to explore this problem.

- Use the left-window to enter the equation. It is important to use
notation. - Plot point A on the x-axis (if you did it right, it should be sky blue)
- Use the reflect tool to reflect A over the y-axis
- Use the left-window to plot the following two points. (x(A),f(x(A)) (x(A’),f(x(A’))
- Switch to the Arrow Tool. If this was done correctly, all 4 points should move when you drag point A.
- Use the Polygon tool to draw a polygon around the 4 points. Poly1 in the left-window shows the area of the rectangle.
- Drag A until the area is at a maximum (keep the rectangle above the x-axis)

**Questions**- What is the maximum area?
- What value of x that maximized the area?
- What is the base length?
- What is the height?
- The area of a rectangle is
. What is the base length in terms of x? What is the height in terms of x? What is the area? - Find the derivative of the area and use this to find when the area of the rectangle has a maximum value.