An open box is constructed from a square 10 -inch piece of...

An open box is constructed from a square 10 -inch piece of cardboard by cutting squares of length $x$ inches out of each corner and folding the sides up. Express the volume of the box as a function of $x,$ and state the domain.

Key Concepts

Function Modeling

This concept involves translating a real-world scenario into a mathematical function. In the context of constructing a box, a function is created by expressing the volume of the box in terms of a variable that represents a key dimension—in this case, the side length of the squares cut from the cardboard. This process requires understanding the relationship between the variable parameter and the physical changes to the object.

Volume Calculation

Volume calculation is a fundamental geometric concept where the volume of a three-dimensional space is determined by multiplying its dimensions. Here, the box's volume is obtained by multiplying its height (the cut-out square's side length) by the area of its altered base, which decreases as the squares are removed from each corner. This leads to a cubic function representing the volume, encapsulating the product of three dimensions that are functions of the variable.

Domain Determination

Determining the domain of a function is crucial, especially when the function models a physical situation. In this problem, the domain is defined by the physical limitations—namely, that the cut-out length must be positive and sufficiently small so that the remaining dimensions of the cardboard are also positive. This ensures that all parts of the box are real and usable, thereby restricting the variable to a specific range.

Transcript

00:01 We've got this 10 inch square piece of cardboard that is 10 by 10, and we want to make it into a box.

00:08 So we're going to be snipping off the corners with these squares of x by x.

00:13 That is, we don't know the length of each side of the square.

00:16 However, the square in each corner is going to be the same as all the other corners.

00:20 So let's express the volume of this box as some function v of x, of course, v for volume.

00:26 So what's this going to be? well, the volume of a box that is a rectangular prism is length times width times height.

00:35 So we need to figure out what the length, the width, and the height are.

00:38 Height is pretty easy.

00:40 Consider folding the box up along these green lines that i'm drawing.

00:44 Then these little flaps here that i'm shading in green would go up and become the sides.

00:50 So the sides would have a height equal to this line here, which is, of course, x.

00:56 So height is going to be x.

00:57 Well, that's good.

00:59 Now then the length and the width of the box, that is this side and this side that i've just traced down in red, those are going to be the same length because this is still going to be square.

01:09 So that's good.

01:10 Now, what are these lengths equal to? well, originally they were of length 10, but now we're subtracting some xes.

01:17 That is, this line here, i'll draw it in blue now, so it really sticks out.

01:22 That's length 10, but we're subtracting segments...

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