Problem Solving: Practice & Approaches

1. Problem Solving: Practice & Approaches Practice solving a variety of problems Strategies for solving problems More Practice

2. General Idea of This Lesson • Programming is like learning a language • You need to learn the vocabulary (keywords), grammar (syntax), and how to use punctuation (symbols) • Problem solving is like learning to cook • A novice chef has a recipe • An master chef can create their own recipe Both tasks require practice!

3. Review: Scientific Problem-Solving Method • Problem Statement • Diagram • Theory • Assumptions • Solution Steps • Identify Results & Verify Accuracy • Computerize the solution • Deduce the algorithm from step 5 • Translate the algorithm to lines of code • Verify Results

4. Example #1: Balancing a fulcrum A 30-kg child and a 20-kg child sit on a 5.00-m long teeter-totter. Where should the fulcrum be placed so the two children balance? (Note: an object is in static equilibrium when all moments balance.) Using the supplied worksheet, solve the problem work on the first couple of steps: On your own With your neighbors What did you get?

5. Example #1: Balancing a fulcrum • Problem Statement: • Givens: m1 = 30 kg m2 = 20 kg L = 5m • Find: Location of the fulcrum • Diagram

6. Example #1: Balancing a fulcrum • Theory Force = Mass * Acceleration (F = m*a) Moment = Force * Distance (M = F*d) MF = 0, The sum of moments about the fulcrum equals zero at static equilibrium • Assumptions Mass of teeter-totter is negligible Earth’s gravitational constant 1 2 3

7. Example #1: Balancing a fulcrum • Solution Steps • Forces: F1 = m1 * g F2 = m2 * g • Moments: M1 = F1 * L1 M2 = -F2 * L2 • Equilibrium: MF = 0 = M1 + M2 0 = F1 * L1 -F2 * L2 0 = m1 * g * L1 -m2 * g* L2 thus: m1 *L1 = m2 *L2 • Get rid of L2 using: L2 = L – L1 m1 *L1 = m2 (L-L1) thus: L1 = m2 * L / (m1 + m2) • L1 = 20*5/(20+30)=100/50=2.00 m

8. Example #1: Balancing a fulcrum • Identify results and verify L1 = 2.00 m Does this make sense? • Units? • Overall Dimension? • Easy to imagine! • Can you rerun the analyses with other givens using Step 5? This is the key to Computer Programming!!

9. Problem Solving Strategies • The trouble with Step 5: “Solution Steps” There can be many approaches to solving the same problem • Creativity is an important component on how we view and approach problems:

10. Creativity • Connect the following 9 dots with four continuous lines without lifting your pencil Sometimes you will need to think outside the box

11. Problem Solving Strategies (Polya, 1945) • Utilize analogies • Flow through a piping system can be modeled with electronics Resistors – Fluid Friction Capacitors – Holdup tanks Batteries – Pumps • Work Auxiliary Problems • Remove some constraints • Generalize the problem Ex: L1= m2* L / (m1 + m2)

12. Problem Solving Strategies (Polya, 1945) • Decompose & Recombine problems • Break the problem into individual components Calculate Cost of Area • Prove the following equation 2 x 2 x 2 x 2 = 16

13. Problem Solving Strategies (Polya, 1945) • Work backwards from the solution Ex: Measure exactly 7 oz. of liquid from an infinitely large container using only a 5 oz. container and an 8 oz. container Solution: • Fill 5 oz container and empty into 8oz • Fill 5 oz container again, then pour to top-off 8oz container (2 oz remaining in 5 oz) • Empty 8 oz and fill with the remaining 2oz from 5oz container • Fill 5 oz container and add it to the 8oz container 8 7? 5

14. Example #2: Fuel tank design A fuel tank is to be constructed that will hold 5 x 105 L. The shape is cylindrical with a hemisphere top and a cylindrical midsection. Costs to construct the cylindrical portion will be $300/m2 of surface area and $400/m2 of surface area of the hemispheres. What is the tank dimension that will result in the lowest dollar cost?

15. Example #2: Fuel tank design • Problem Statement: Givens: CostHemisphere = $300/m2 CostCylinder = $400/m2 VolumeTank = 500,000 L Find: Size for minimum cost • Diagram R H

16. Example #2: Fuel tank design • Theory Volume Cylinder : Volume Hemisphere: Surface Area Cylinder: Surface Area Hemisphere: • Assumptions No dead air space Construction cost independent of size Other costs do not change with tank dimensions Thickness of walls is negligible Bottom portion of tank is free.  1 2 3 4

17. Example #2: Fuel tank design • Solution Steps 1. Substitute equations Solve for H with respect to R 2. Substitute equations • Solve for Cost with • respect to radius

18. Example #2: Fuel tank design • Solution Steps Calculate minimum cost with respect to radius MATLAB can be used to calculate minimum cost • Plot the data - plot(R,Cost) • Identify minimum for an array of costs - min(Cost) • Numerical Methods (Iterative solutions)

19. Example #2: Fuel tank design • Solution Steps Solve for H with identified R where • H = 3.03m R H

20. Example #2: Fuel tank design • Identify results and Verify Take the derivative of the cost function Does this make sense? • Units? • Overall Dimension? • Can you rerun the analyses with other givens using Step 5? 5m 3m 5VTank = 500,000L

21. Wrapping Up • Utilize the 7 step process before you begin programming • Be clear about your approach • Think creatively • Use a couple of strategies when understanding a problem • Practice! • Use MATLAB to make your life easier

22. Try it yourself • What if the fuel tank had two hemispheres? A fuel tank is to be constructed that will hold 5 x 105 L. The shape is cylindrical with a hemisphere top, a hemisphere base and, and a cylindrical midsection. Costs to construct the cylindrical portion will be $250/m2 of surface area and $300/m2 of surface area of the hemispheres. What is the tank dimension that will result in the lowest dollar cost? R H