SOLUTION: A rectangular prism has a surface area of 256 sq in. What is the maximum volume it can hold?

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Question 399060: A rectangular prism has a surface area of 256 sq in. What is the maximum volume it can hold?
Answer by richard1234(7193) About Me  (Show Source):
You can put this solution on YOUR website!
Suppose the dimensions of the prism are L, W, H. Then,

2%28LW+%2B+WH+%2B+LH%29+=+256 --> LW+%2B+WH+%2B+LH+=+128

By the AM-GM inequality (arithmetic mean - geometric mean inequality),

%28LW+%2B+WH+%2B+LH%29%2F3+%3E=+root%283%2C+L%5E2W%5E2H%5E2%29

128%2F3+%3E=+root%283%2C+L%5E2W%5E2H%5E2%29 Cubing both sides,

128%5E3%2F3%5E3+%3E=+L%5E2W%5E2H%5E2

sqrt%28128%5E3%2F3%5E3%29+%3E=+LWH, this value is the maximum volume of the prism, which occurs when the solid is a cube.