Writing Volume as a Function of Height for an Open Box

M83

1. The problem statement, all variables and given/known data

An open box of maximum volume is to be made from a square piece of material 24 centimeters on a side (24-2x) by cutting equal squares from the corners and turning up the sides. The table shows the volumes V (in cubic centimeters) of the box for various heights, x (in centimeters).

(x, V): (1,484), (2,800), (3,972), (4,1024), (5,980), (6,864)

If V is a function of x, write the function and determine its domain.



3. The attempt at a solution

I'm completely stuck on this. I tried recreating the table values by using the volume of a cube formula, but that didn't work. If anyone could give me a nudge in the right direction that would be helpful, thanks.

SammyS

What cubic function do you get for the volume of the box ?

HallsofIvy

The problem tells you that the base is a square that has side length 24- 2x. What is the area of the base? How do you go from "area of base" to "volume"?

M83

For a square the area would be the square of the side length.

A= (24-2x)(24-2x)
= 576-48x-48x+4x²
= 4x²-96x+576

Would you cube the side length?

Ray Vickson

Why on Earth would you do that? If I have a box whose base has area 10 m² and whose height (= sides) are 2 m, what is the volume (in units of m³)?

RGV

eumyang

That's because you don't have a cube to begin with, you have a rectangular prism ("box"). You do know that the volume of a rectangular prism is [itex]V = lwh[/itex] (l = length, w = width, h = height), right?