Finding a Function for VolumeA carpenter has been asked to build an open box with a square base. The sides of the box will cost 4 dollars per square metre and the base will cost 5 dollars per square metre. 60 dollars is available to build the box. If the dimensions of the box are x, the side of the base, and y, the height of the box, find a function for the volume in terems of one of the variables.

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Step 1You want the volume of the box. That’s the area of the base times the height, so   V  =      x    2    y; the problem with this, of course, is that it contains both x and y, and you need to find an expression that contains only one of them. Presumably that’s what the other information in the problem is good for. Let’s see what we can dig out of it.Step 2There’s only one base, so let’s start there. It will cost 5 dollars a square metre, so its total cost in dollars will be 5 times its area. Each side of the square base is x metres long, so the area of the base is       x    2   square metres, and the cost of the base is   5      x    2   dollars. You can use similar reasoning to get an expression for the cost of one side, which you can then multiply by 4 to get the total cost of the four upright sides. Adding this subtotal to   5      x    2   will give you the total cost of the entire box, though your expression will again contain both x and y.Fortunately, there’s one piece of information that we’ve not used yet: the total cost is 60 dollars. Set your expression for total cost equal to 60; that will give you an equation involving both x and y. You can solve that equation for one of the variables in terms of the other.

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