Optimization - Rectangle Inscribed in a Parabola
A rectangle is inscribed between the `x`-axis and a downward-opening parabola, as shown above.
The parabola is described by the equation `y = -ax^2 + 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!
- 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.