More optimization problems.

1. This box isn't big enough
for the both of us!
Optimization to the rescue!
Bigger Box by flicker user brainwise

2. Problem: Find the rectangle with the maximum area which can be
inscribed in a semicircle.

3. A rectangle is bounded by the x-axis and the semicircle .
What length and width should the rectangle have so that its area is a
maximum?

4. An open box is to be made from a square piece of material, 24
inches on a side by cutting equal squares from the corners and
turning up the sides. What size squares should be cut out to
maximize the volume of the box?