A rectangular storage container with an open top has a volume of20m³.The length of its base is twice its width. Material for the base costs $7per square meter; material for the sides costs $5per square meter. Express the cost of materials as a function of the width of the base.

A rectangular prism is shown. The length is labeled 2w, the width is labeledw, and the height is labeledh.

Solution

We draw a diagram as in the figure and introduce notation by lettingwand 2wbe the width and length of the base, respectively, andhbe the height.

The area of the base is(2w)w= 2w²,

so the cost, in dollars, of the material for the base is(2w²).

Two of the sides have areawhand the other two have area 2wh, so the cost of the material for the sides is[2(wh) + 2(2wh)].

The total cost is therefore

C=

(2w²) +

[2(wh) + 2(2wh)]

=

w²+

wh.

To expressCas a function ofwalone, we need to eliminatehand we do so by using the fact that the volume is20m³.Thus

w(2w)h=

which gives

h=

2w²

=

w²

.

Substituting this into the expression forC, we have

C=

w²+30w

w²

=

w²+

w

.

Therefore the equation

C(w) =

w²+

w

expressesCas a function ofw(w> 0).