The location is near Pittsburgh on the bank of the Monongehela River not far from the old Westinghouse East Pittsburgh plant. You've got the job of shipping V cubic feet of old steel shavings across the river by barge. To do this, you'll make a box to put the shavings in for the barge shipment. Put
x = length of the box in feet,
y = width of the box in feet, and
z = height of the box in feet.
The material for the box consists of five rectangular pieces
The bottom has area x y; the material for it costs Sa per square foot. So in dollars,
bottom cost = a x y.
Another two pieces (the sides) have area xz cach: these cost Sb per square foot. So in dollars
side cost = 2 b x z
And the other two pieces (the ends) have area y z these cost Sc per square foot. So in dollars,
end cost 2 c y z
The tugboat captain tells us that she will charged dollars for each trip. You have V cubic feet of shavings, so you'll need trips. So in dollars,
Transportation cost = d V/x y z
The bottom line is
overall cost bottom cost + side cost and cost transportation cost
What should the ratios
bottom cost de contend cost transportation cost
be if the overall cost is at a minimum?
Give a formula for the minimum cost and the corresponding optimal dimensions, x, y, and in terms of a, b, c, d, and h
How does the minimum cost change if the captain changes her mind and charges 2 d dollars per trip? d , and V.