79. A square piece of cardboard measures 12 inches on each...

79. A square piece of cardboard measures 12 inches on each side. Four squares, each having a side of x inches, are cut and removed from each of the four corners of the square piece of cardboard. The sides are then folded up along the dashed lines to form a box with no top. a) Find the volume of the box as a function of x, the measure of the side of each square cut from the four corner of the original piece of cardboard. Multiply to place your answer in standard polynomial form, simplifying your answer as much as possible. b) Use the resulting polynomial to determine the volume of the box if squares of length 1.25 inches are cut from each corner of the original piece of cardboard. Round your answer to the nearest cubic inch.

Transcript

00:01 In this problem, we have a square sheet of cardboard, and we're cutting out squares from each of the corners.

00:09 We're upturning the edges so that we can get an open -face box.

00:14 So we have the diagrams here for you.

00:17 The one on the left will turn into the one on the right.

00:22 So the first question is asking us to find the volume of this open -faced box as a function of x, where x is the length of each of these little squares that we had cut out.

00:34 We need to write this as a standard polynomial.

00:40 So, okay, let's do that.

00:42 Volume of a box, volume of a rectangular box in particular, is length times width times height.

00:51 So we already have the height.

00:53 It's right here.

00:54 This is our height.

00:55 It's x.

00:55 We don't know what these are just yet.

01:00 So it's going to be coming from this picture right here.

01:03 We know that the entire thing used to be 12, but some parts got taken away from it.

01:09 So if i draw in or write an x here and x here, that's how much was taken away from this entire length of 12.

01:18 So what's left over? that's the question.

01:21 Well, if this is x, this is x times that by 2, we're taking away 2x from this 12.

01:28 So what's left is 12 minus 2x.

01:33 Let me write that in a different color.

01:35 So it stands out a little better.

01:36 12 minus 2x.

01:39 So that's going to be this length or width, depending on what you want to call it.

01:45 And in a very similar fashion, since we do have squares here, and this is also an x, and this is also an x, and this is also the whole thing is 12.

01:57 In between, we're going to have the same thing.

02:00 12 minus 2x.

02:05 So now we have the length, width, and height that we need in order to find the volume of the box.

02:12 So let's go ahead and start writing that out.

02:16 Volume.

02:16 So v for volume.

02:18 We're going to take x times 12 minus 2x times itself, which i'm just going to write it next to it right here.

02:29 Okay, so we need to have in standard polynomial form, meaning we need to multiply.

02:33 Everything out.

02:34 So let's do that.

02:35 I'm going to ignore this x for a while and just work with these two binomials.

02:41 I got to do foil.

02:44 So i have 144 minus 24x minus another 24x.

02:56 And then, oops, did i do that a little too quickly? so 144 minus 24x...

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