1. The Boat Challenge

2. Materials 1) One 8.5 x 11 inch sheet of laminated paper 2) Scotch tape 3) Scissors 4) Ruler

3. Design/Build Boats  Build a rectangular boat that will hold the most weight without sinking.

4. Requirements  Boat must have a name.  Boat must have a flag.  Boat must be completed in 20 minutes.

5. Test Boats  Place boats in the tub of water.  Apply weights until the boat sinks.  Count the total mass the boat was able to carry (not including the mass that caused the boat to sink) Note: Most boats carry several hundred grams. Be sure to use heavy mass first.

6. Volume … is the quantity of interest  Buoyancy: A boat floats when its weight is displaced in water.  Therefore – the best boat will displace the most water.

7. Design the boat with Maximum Volume  Write a function for volume.  Maximize the function.

8. Functions for Volume  f(h) = (11 – 2h)(8.5 – 2h)(h)  f(h) = 4h 3 – 39h 2 + 93.5h

10. Maximize using Calculus  1 st Derivative: f’(h) = 12h 2 – 78h + 93.5  Set f’(h) = 0 and solve for h 12h 2 – 78h + 93.5 = 0 12h 2 – 78h + 93.5 = 0  h = 78 ± (78 2 – (4)(12)(93.5)) 0.5 (2)(12) (2)(12) h ≈ 4.915 inches & 1.585 inches

11. Which is my solution? h ≈ 4.915 or 1.585 inches  2 nd Derivative: f’’(h) = 24h – 78  f’’(4.915) = 39.96 and f’’(1.585) = -39.96  f(1.585) ≈ 66.158 inches cubed

12. Optimization  One of the most powerful tools math- ematics offers industry is optimi- zation. The ability to predict and design the best “specimen” given various goals and constraints is lucrative.

13. New Challenge  Given an 8.5 x 11 inch sheet of paper, build the best “canoe.”  “Canoe” = triangular ends and one crease down the middle.

14. New Challenge waste

15.  “Canoe” = triangular ends folded up at 90 ˚ and one crease down the middle. waste