Optimization Examples

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• Questions?
• Optimization
  • Examples
    • Do manufacturers of buckets actually use dimensions that come close to minimizing the amount of material needed to make the bucket?
      • Bucket = cylinder
      • Surface area proportional to amount of plastic and cost
        • = 2πrh + πr²
      • Volume V = πr²h gives
        • h = V/(πr²)
      • So A = 2πrV/(πr²) + πr²
        • = 2V/r + πr²
          
      • Optimal r = (V/π)^1/3
      • V = 2.5 gallons = 577.5 cubic inches
        

        > V = 2.5 * 231
        > V
        [1] 577.5
        > r = (V/pi)^(1/3)
        > r
        [1] 5.685919
        > h = V / (pi*r^2)
        > h
        [1] 5.685919

        • Optimal r = 5.68, h = 5.68
        • Actual r = 4.5, h = 10
    • Optimal route for getting lawn mower back to shed.
      
      • Differentiate equation for time and find x that makes it 0
        
      • Time for this x is less than times at endpoints of x interval, so optimal route crosses lawn to point 57.7 feet from end of driveway and then follows driveway
• Next
  • Introduction to antiderivatives aka integrals
    • Formalities
      • Notation
        
      • Rules (I added 2 re trigonometric antiderivatives that we didn’t mention in class)
        
  • Antiderivative as a sum of function values
    
  • Read section 5.1

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