(NOTE: In all the examples below, I pressed CTRL-9 [which gives "( )" and marks the equation for GC as a function defintiion] instead of SHIFT-9 [which gives simply the character "("] when defining a function. See "Help>Graphing Calculator Help>How to Enter Formulas>How to Type Functions".

We have a building that will appear as in this diagram. The length of all four north-south walls is labeled "w". The width of the building is the length of these walls. Therefore,

function(W,w)=w

We first begin by defining the relationships between the "perimeter" (and area) and the length and width of the building. We see that the total amount of wall material, 2000 feet, will be used by 4 north-south segments, each of length w, and 2 east-west segments, each of length l (el, not one). So, P = 4w + 2l, and P = 2000. so the building's length expressed as a function of its width is, where w is the width of the building (and each of the two vertical walls), and l is the length of the building. Therefore,

function(L,w)=(2000-(4*w))/2

This says that the building's length (as measured by the length of its two "east-west" walls) is the length of wall that can be made with the leftover after making its 4 "north-south" walls. It expresses the invariant relationship between w, the building's width as measured by the length of a north-south wall), and L(w), the length of an east-west wall given that its north-south walls are w feet long.

You can see that when the buiding is 0 feet wide it is 1000 feet long. As the width increases the length decreases, until the length is 0 when the width is 500. (Of course, the width wouldnt' really be zero, because you would have four rows of material stacked against each other, and the "building", though having no floor space, would have a width.)

function(A,w)=function(W,w)*function(L,w)

This says that the building's floor area, given that a north-south wall is of length w, is the product of the building's width and the building's length.

y=function(A,x),0<x<500

The graph shows that the building's floor area increases as the building's width increases, until the width is approxximately 250 feet, after which the floor area diminishes. Clicking on the graph allows us to use the trace feature. It seems that a width of 250 (and therefore a length of 500) yields the largest area.

Graph of the formula

This file was created by Graphing Calculator 3.0.
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