Question #eb797

Let us set up the following variables:

`{(w, "Width of Building (yards)"), (l, "Length of Building (yards)"), (h, "Height of Building (yards)"), (V, "Volume of the Building (cubic yards)") :}`

We want to vary the dimensions such that we maximise `V`, i.e. find a critical point of `(dV)/(dw)` that is a maximum, so we to find a V as a function of one independent variable function `V=V(w)`

Then the volume is:

`V=wlh`

The roof costs $`d` per square foot so the cost of the roof per square yard is `3*3*d=9d`. Therefore the material costs of the building, and the Surface Areas are given by:

`{: ( "Component", "Surface Area sq yards", "$ cost per sq yard" ), ( "Foundation", wl, 4*("walls")=72d), ( "Walls", 2wh+2lh, 2*("roof")=18d), ( "Roof",wl, 9d) :}`

So the Total material cost, `D`, is given by:

`\ \ \ \ \ D = (wl)(9d) + (2wh+2lh)(18d) + (wl)(72d)`
`:. D = 9wld + 36whd+36lhd + 72wld`
`:. D = 81wld + 36whd+36lhd`
`:. 9wl + 4wh+4lh = D/(9d)`
`:. 9wl + 4V/l+4V/w = D/(9d)` (`D` and `d` are constants)

And so we have reduced the problem to the Volume being a function of two variables `V=V(w,l)`. I could go on and look at the partial derivatives, `(partial V)/(partial l)` and `(partial V)/(partial w)` but I suspect the question is not that "deep" and instead another constraint is missing.