This is much more fun:

You have 100 ft of fence and want to make a rectangular pen to house your favorite goat, (definitely let them name the goat). Of course, you want the goat to be happy. (Then I talk about the fact that the goat will want lots of room to frolic - the word "frolic" slays me for some reason.) Find the dimensions to maximize the Goat-Frolic-Space.

Before setting up the problem, I really get into the different types of rectangular pens can be made. What if we made the pen 1 ft by 48 ft?

Then I pretend I'm the goat in the pen. I shuffle forward, hit the pretend wall of the pen and baaaa... Then, since there's no room to turn around in there, I shuffle backwards, hit the pretend wall of the pen again and baaaa. Yes, it's extremely silly. I admit it! I do a mean impression of a billy goat though.

While solving, we use the special GFS (Goat-Frolic-Space) function, GFS(x).

By the way, after we solve the problem - and get a square, of course - I explain to them that the rectangle really isn't the best shape for the pen... If you take the same amount of fence and make it into a circle (using the circumference of 100 ft, you can find the radius and then compute the area), you'd be amazed how much larger the Goat-Frolic-Space would be!

How about the boring maximize-the-volume-of-the-box problem?

Try this:

You've just bought a new pet grasshopper and named him (let them name him). You need to build him a house - this will be a topless box ... (yadda yadda yadda about cutting the corners out of a square piece of cardboad and bending them up - I'll let you fill in the math) (Now, I explain that the grasshopper will want room to jump up and down and run around back and forth.) You want your new pet to be happy! We need to maximize the psychological well-being of the grasshopper (i.e. the volume).

These are just goofy twists on the old problems, but they really make them a lot more fun for everyone.

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