Optimization word problem - minimizing surface area to find least expensive tank

[cahsuhdee]

Homework Statement

A tank with a rectangular base and rectangular sides is to be open at the top. It is to be constructed so that its width is 4 meters and its volume is 36 cubic meters. If building the tank covers $10 per square meter for the base and $5 per square meter for the sides, what is the cost of the least expensive tank?

Homework Equations

Surface Area of a Rectangular Prism 2(LW) + 2(LW) of other set of sides + LW of base
Volume = LWH

The Attempt at a Solution

First thing.. come up with the primary Equation... and since I'm trying to minimize Surface Area... SA = 8H (smaller sides) + 8LH (larger sides) + 16L (base)

Secondary Equation, in order to manipulate the variables so I can plug them into the primary equation... Volume = LWH = 36

And then... after plugging whatever expression I get for the variable into the primary eq... I know I do the derivative.. set it equal to 0.. then solve.. or at least that's what we HAVE been doing for previous problems.. But before it was always something like.. "Find the dimensions of blah which would maximize the volume" .. and they never had this many variables.. so really I'm not sure what I'm doing, after the mere realization that somehow I use the volume to apply it to the surface area, then multiply it by cost to find how much it'd be. I'd be thankful for any sort of help!

 

[hotvette]

I'm trying to minimize Surface Area

First thing you need is to be clear on what you are to minimize. It isn't surface area. Read the problem statement again.

Write an equation for what it is you are trying to minimize and use the given information to simplify the equation to a single variable. Once you have that, set the derivative equal to zero and solve for the value of the variable.