# A metal box with a fixed volume of 8 cubic meters is to be constructed using gold-plating for the… 1 answer below »

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A metal box with a fixed volume of 8 cubic meters is to be constructed using gold-plating for the top, silver-plating for the bottom, and copper-plating for the sides. Gold plating costs $120 per square meter, silver-plating costs $40 per square meter. and copper-plating costs $10 per square meter. Suppose we want to minimize the total cost of coating such a box.

1) Write the cost function to be minimized. How many input and output variables does the cost function have? (1 point) 2) Are the inputs subject to a constraint equation? If yes, write the constraint equation. (1 point) 3) Use the methods from section 14.7 to find the dimensions of the metal box that minimize the cost of the coating, and then find the minimum cost of coating the metal box. (4 points)

Hint: In order to use the methods of section 14.7, your cost function must have only two input variables. Use the constraint equation to eliminate one of the input variables from the cost function, say z.

4) Use the method of Lagrange Multipliers from section 14.8 to find the dimensions of the metal box that minimize the cost of the coating, and then find the minimum cost of coating the metal box. (4 points)

Verify you get the same answers to #3 and #4.

Hint: To get started with solving the Lagrange System. use the addition/elimination method from algebra to eliminate the terms from the first two equations.

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