A rectangle is inscribed between the x-axis and a downward-opening parabola, as shown above.
The parabola is described by the equation y = −ax2 + b where both a and b are positive.
You can reshape the rectangle by dragging the blue point at its lower-right corner.
Note: x is the distance from the origin to the lower-right corner of the rectangle; x is
not the length of the base of the rectangle!
Explore
- Let a = 1 and b = 7. What value of x maximizes the area of the rectangle?
- Let a = 1 and b = 7. What value of x maximizes the perimeter of the rectangle?
- Repeat the above two problems for a and b in general.
- From the previous exercise you can see that the x value where the perimeter
is maximized depends only on the parameter a. Describe all parabolas that have
an inscribed rectangle of maximum perimeter at x = 1.
- Occasionally it happens that for a given parabola the same value of x maximizes
the area and the perimeter of the rectangle. If a parabola has this property, what is
the relationship between a and b? Verify you findings by trying a few
examples with the applet.