Suppose you want to build a fish tank in the shape of a right rectangular box with square base and no top which will hold 6 cubic feet of water. the glass for the sides costs $1 per square foot, and the metal for the bottom costs $1.50 per square foot. what dimensions for the tank will minimize the cost?

You must develop a cost function C(x) and then minimize its value.
How much dwill the glass cost?  It's $1 per sq ft, and the total area of the glass is 4(xh), where x is the length of one side of the base and h is the height of the tank.  The area of the metal bottom is x^2, which we must multiply by $1.50 per sq ft. 

This cost function will look like this:  C(x) = 4($1/ft^2)xh + ($1.50/ft^2)x^2

but we know that (x^2)h= 6 cu ft, or h = (6 cu ft) / (x^2).  Subst. this last result into the C(x) equation, immediately above:

C(x) = 4($1/ft^2)x[6 ft^3 / x^2] + ($1.50/ft^2)x^2

Let's focus on the numerical values and ditch the units of measurement for now:

C(x) = 4x(4/x^2) + 1.50x^2, or

C(x) = 16/x + 1.5x^2

Differentiate this with respect to x:

C '(x) = -16 / x^2 + 3 x

Set this equal to 0 and solve for x:    -16/x^2 = -3x, or 16 = 3x^3

Then x^3 = 16/3, and x = 5 1/3 ft.  We already have the formula  

(x^2)h= 6 cu ft, so if x = 5 1/3, or 16/3, then (16/3)^2 h = 6, or

h = 6 / [16/3]^2.

h = 6 (9/256) = 0.21 ft.  While possible, this h = 0.21 ft seems quite unlikely.

Please work through this problem yourself, making sure you understand each step.  If questions arise, or if you find an error in my approach, please let me know.

Once again:
1.  Write a formula for the total cost of the material used:  4 sides of dimensions xh each, plus 1 bottom, of dimensions x^2.  Include the unit prices:  $1 per square foot for the sides and $1.50 per square foot for the bottom.
2.  Differentiate C(x) with respect to x.
3.  Set C '(x) = 0 and solve for the critical value(s).
4.  Calculate h from your value for x.
5.  Write the dimensions of the tank:  bottom:  x^2; height:  h

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