# Question #eb797

##### 1 Answer

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

Then the volume is:

# V=wlh #

The roof costs $

# {: ( "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 = (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