Mathematics

Problem

A manufacturer must fabricate an open-top rectangular box with volume exactly

32{,}000\text{ cm}^3

. The material for the base costs \

0.05 per square centimeter, while the material for the four sides costs

0.02 per square centimeter. Answer each part, showing all reasoning and algebraic justification. (a) Write the total cost

C

of materials in terms of the base length

x\text{ cm}

, base width

y\text{ cm}

, and height

h\text{ cm}

. (b) Use the volume condition to eliminate one variable and obtain

C

as a function of two variables. (c) Further reduce

C

to a single-variable function and determine its derivative. (d) Find the critical point(s) of

C

, and determine the dimensions that minimize the total cost. (e) Demonstrate that the cost obtained in part (d) is the global minimum over the feasible domain, and state the minimum possible material cost to the nearest cent.

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