(1) German mathematician Johannes Regiomontanus 
       (1436 –1 476) posed the following question:
       From what position along a horizontal line can 
       a statue best be viewed?  If the spectator is 
       too close, the statue will appear heavily 
       foreshortened, thus distorting its size.  If 
       the spectator is too far away, it will simply 
       be too small to see.  An optimal distance for 
       viewing the statue must exist.

       For example, the height of the Statue of Liberty is 
       305 feet from the ground to the tip of the torch. 
       If you are on a boat in the harbor and the deck is 
       10 feet below base of the statue, how far away
       should you be to view the statue at its optimal 
       angle?  What is the optimal angle?

       Answers: 56.12 feet, angle is 69.79 degrees


   (2) Given an isosceles triangle with equal sides of 3 
       inches in length.  Determine the measure of the 
       angle between the two equal sides which results in 
       the largest area.     

       Answer: 90 degrees

    
   (3) Construct a window in the shape of a semi-circle 
       over a rectangle (like many stained glass windows in 
       churches).  If the distance around the outside of 
       the window is 12 feet, what dimensions will result 
       in the rectangle having the largest possible area?

       Answers:  The rectangle's dimensions would be 2.33 
       feet by 3 feet, so the area would be 6.99 square 
       feet.

   (4) Ladder in the hall Problem
       A non-folding ladder is to be taken around a corner 
       where 2 hallways intersect at right angles.  One 
       hall is 7' wide and the other hall is 5' wide.  What 
       is the maximum length that the ladder can be so it 
       will pass around the corner? 

       Answer:  16.89 feet

   (5) During the Winter break, Matt E. Matics is enroute 
       to his sister's house.  Two policemen are stationed 
       two miles apart along the road, which has a posted 
       speed limit of 55 mph.  The first clocks him at 50 
       mph as he passes, and the second policeman clocks 
       him at 55 mph, but pulls him over anyway.  When he 
       asks why he got pulled over, the policeman responds, 
       "It took you 90 seconds to travel 2 miles."  Why is 
       he guilty of speeding according to the Mean Value 
       Theorem?

       Answer: His average rate is 80 mph.  There must have
       been at least one point over the interval where the 
       instantaneous rate of change (velocity) must equal 
       the average rate of change (average speed).  
       Therefore, old lead foot Matt must have traveled 80 
       mph at least once, and the policeman can ticket him.

   (6) The practice of shooting bullets into the air – for 
       whatever purpose – is extremely dangerous.  Assuming 
       that a hunting rifle discharges a bullet with an 
       initial velocity of 3000 ft/sec from a height of 6 
       feet, answer the following questions:
       (A) How high will the bullet travel at its peak?
       (B) At what speed will the bullet be traveling when 
           it slams into the ground, assuming that it hits 
           nothing in its path?   
       
       Answers: (A) 140,631 feet (it takes 93.75 seconds to 
                    reach the maximum height)   
                (B) 3000.064 ft/sec 

   (7) To celebrate our 40th Anniversary in 2008, I am
       commissioning the construction of a four-inch tall 
       box made of precious metals to give to my wife.  
       The jewelry box will have rectangular sides and an 
       open top.  The longer sides of the box will be made 
       of gold, at a cost of $300 per square inch; the 
       shorter sides will be made of platinum, at a cost 
       of $550 per square inch.  The bottom will be made 
       of plywood, at a cost of 2 cents per square inch.  
       What dimensions provide me with the lowest cost if 
       I am adamant that the box have a volume of 50 cubic 
       inches?
      
       Answer:  C = 2400x + 4400y + xy(.02)
                Dimensions are 4.787 in x 2.611 in x 4 in. 
                (The lowest cost for the box will be 
                 $23,000 – do I still need to buy a card?) 
         
   (8) A builder is purchasing a rectangular plot of land 
       with frontage on a road for the purpose of 
       constructing a rectangular warehouse.  Its floor 
       area must be 300,000 square feet.  Local building 
       codes require that the building be set back 40 feet 
       from the road and that there be empty buffer strips 
       of land 25 feet wide on the sides and 20 feet in the 
       back.  Find the overall dimensions of the parcel of 
       land and building which will minimize the area of 
       the land parcel that the builder must buy.

       

       Answer:  Dimensions of the building: 500' x 600'
                Dimensions of the land: 550' x 660'