Question

(15 points - 3 points each) A rectangular storage container with open top is designed to have a volume of 10 cubic meters. The length of its base is twice its width. Assume the material for the base costs 10 dollars per square meter, and the sides 6 dollars per square meter. Assume also you wish to minimize the cost of a container of that volume. (a) Draw a sketch of the situation. (b) Write down the function to be optimized in terms of height and width. (C) Express volume as a function of height and width. (d) Use the constraint to express the cost as a function of just one variable. (e) What is the cost of the cheapest such container?

Answer 1

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volune VE X. 2x.y = 10 m3 ly xawidth ya height 2x²y = 10 x2y=5 24 x 95 x² 0 cost for base = 10(2x.x)= 2012 Cost for sides = 6 (2x) (y) = 12xy cost for sides = 6(%) (y) = 6xy Total cost ((x,y) = 20x²+1884, v=2x²y = 10 3i y=5 x2 (@ ((x) = 20x² 4 1885 20x?+ 90 x2 x e To minimize cost c/ex) = 400-90 X2 40x3_go x2 the Go = 914 ala x=(943 १० ((*)= 2014 2) 113 103-024 $