Question 1137049
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The volume equation is  


    x^2*h = 108.     (1)


The surface area expression for the open box is  


    A(x,h) = x^2 + 4xh.     (2)


So, you need to find dimensions which minimize the function A(x,h) (2)  under the condition (1).


To solve the problem, express h = {{{108/x^2}}} from (1) and substitute it into (2), making A function of only one variable x:


    A(x) = x^2 + 4x*{{{108/x^2}}} = x^2 + {{{432/x}}}.    (3)


Now you have this function A(x) of one variable x, and you should find its minimum.


Differentiate; equate the derivative to zero


    A'(x) = 2x - {{{432/x^2}}} = 0

and get


    2x^3 - 432 = 0  ====>  x^3 = 432/2 = 216  ====>  x = {{{root(3,216)}}} = 6.


<U>Answer</U>.  x= 6;  h = {{{108/x^2}}} = {{{108/6^2}}} = {{{108/36}}} = 3.
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Solved.