Question 1155393
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<pre>

Let x be the size of the squared base edge and h be the height.


Then the volume is  x^2*h = 100  cubic inches 

and the lateral surface area is 4*xh square inches.


The total cost is  C(x,y) = 5x^2 + 2*4*xh = 5x^2 + 8xh.


So we need to minimize  C(x,y) = 5x^2 + 8xh  under restriction x^2*h = 100.

We then express h = {{{100/x^2}}} and substitute it into the expression for C(x,y).


We then get

    C(x,y) = {{{5*x^2}}} + {{{8x*(100/x^2)}}} = {{{5x^2}}} + {{{800/x}}},

and we need to minimize this function.


Find the derivative and equate it to zero

    10x - {{{800/x^2}}} = 0.


From this equation, find x

    10*x^3 = 800

    x^3 = {{{800/10}}} = 80.

    x = {{{root(3,80)}}} = 4.309  inches.

Then h = {{{100/x^2}}} = {{{100/4.309^2}}} = 5.386 inches.


<U>ANSWER</U>.  The base size is 4.309 inches;  the height is 5.386 inches.
</pre>

Solved.


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