Donkey Kong wants to make a 250 mL fruit smoothie. Being an ape, naturally he would like a large proportion of the smoothie to be bananas—at least 100 mL but not more than 150 mL. There should be at least 10 mL each of apples, cherries, and dates. However, for proper taste, there should be twice the amount of apples as cherries. The prices at the local supermarket are as follows:

Donkey Kong wants to spend as few coins as possible on his smoothie.

(a) What are the design variables in this optimization problem?

(b) Translate the written description of the problem into a formal optimization problem statement (see Chapter 9 in the textbook). Label each expression as an objective function, inequality constraint, equality constraint, or bound on a design variable.

(c) Using any strategy you wish, find the values of the design variables that minimize the objective function while satisfying the constraints (i.e., solve the optimization problem). How many coins will the smoothie cost? (If you decide to use Matlab, see the linprog function; if you prefer Python, see scipy.optimize. Does the answer returned by the software make sense?)

Bowser tripped over a Goomba while gardening and hurt his ankle. He has asked Dr. Mario to design a device that attaches to his leg, with a spring at the ankle to help him walk. Unfortunately, Dr. Mario is a general practitioner and has limited knowledge of orthoses, so he has decided to use an optimizer to help design the device.

(a) Describe a strategy to help Bowser walk, using language and referring to material that has been covered in the course. State which structures you are assuming Bowser has injured. You can assume Bowser's lower-limb anatomy is similar to that of a human.

(b) Propose (but do not solve) an optimization problem that Dr. Mario can use to design the device you described in part (a). Explain each expression in your optimization problem.