Optimization problems are designed to extend the extrema tasks we did earlier to real-world situations. General strategies for finding extrema of contextual problems:
Cereal box. Consider a rectangular side of a cereal box. The marketing department wants 90 square inches of area for the front of the box. What is the minimum perimeter?
Three-sided pen. A farmer buys 60 feet of fence to build a rectangular pen off his barn. Thus the fencing will be used for three sides of the fence and the barn will be the fourth side. What are the dimensions for the pen with the largest area?
Two gardens. An eccentric gardener wants to create two rectangular gardens next to one another with a total area of 408 3/8 square feet. The gardens will be surrounded by fence and will share the fence between them. What is the minimum length of fencing needed to create these gardens?
Remove corners. An open-topped box to be constructed from a square piece of cardboard, 3 meters wide, by cutting a square from each of the four corners and bending up the sides. What is the maximum volume of the box described above? (Geogebra visual)
Challenge. Inscribe a rectangle in a semicircular lamina (i.e. half-moon shape) with radius r. What is the area of the largest rectangle that can be inscribed into this semicircle? (Desmos visual)
Hint: When creating formulas, remember that r is constant in this problem!