Constructing a Box A rectangular piece of cardboard, whose area is 216 square centimeters, is ma

Constructing a Box A rectangular piece of cardboard, whose...

Key Concepts

Area Constraint of the Original Cardboard

The original piece of cardboard has a fixed area, which sets a constraint on the possible dimensions of the rectangle. The product of the length and width of the cardboard must equal this predetermined area, thereby linking the dimensions of the starting material to those of the resulting box.

Volume Calculation of a Box

The volume of a box is determined by multiplying the base area—the product of its length and width—by its height. In the context of an open box constructed from a rectangular piece of cardboard, the height is typically equal to the side of the square that was removed, and the base dimensions are adjusted for the cuts made.

Algebraic Equation Formulation and System Solving

Solving such problems typically requires setting up equations that represent both the area of the original material and the volume of the constructed box. By expressing the modified dimensions in terms of the original dimensions and the size of the cuts, one obtains a system of equations. These equations can then be solved using algebraic methods to determine the original dimensions needed to achieve the desired box volume.

Open Box Construction

This concept involves modifying a flat, rectangular sheet into an open-topped box by removing identical squares from each corner and folding up the resulting flaps. It is a common method used in optimization problems and geometry to transform two-dimensional materials into three-dimensional objects.

Dimensional Reduction Due to Corner Cuts

When squares are cut from each corner of the rectangular sheet, the length and width of the base of the box are both reduced by twice the side length of the cut-out squares. This reduction is a crucial factor in determining the final dimensions of the box and directly affects both the base area and the overall volume.

Transcript

00:01 All right, so in this problem, we are given a rectangle, and that rectangle has dimensions that we don't know yet.

00:08 Our goal is to find those dimensions.

00:10 We do know that they multiply to 216 square centimeters.

00:16 We also know, so that's x times y equals 216.

00:20 We also know that if you take out the corners two centimeters along each side and then take out the corner that corresponds to that, and you fold up the edges, then you fold up the edges, then you, you end up with something that looks like this, where it's x long, y, wide, and two high.

00:41 So the volume is two, and then this is actually not x because we lost two here and two here.

00:48 So this is x minus four, so two times x minus four.

00:53 And then same thing with the y.

00:54 This is actually isn't y.

00:55 It's y minus the two and minus the two.

00:58 So y minus four.

01:02 And this expression equals 224 cubic centimeters because this is a volume that we want the box to create.

01:10 So next what i'm going to do is rewrite both of the equations in a slightly simpler way.

01:16 First one is i'm going to solve this one for one of the variables.

01:20 It doesn't matter which one.

01:21 I'm going to solve for x.

01:25 So divide both sides by y and you get x is equal to 216 over y.

01:30 For the other equation, what i'm going to do is divide both sides by two so that i can simplify my numbers a little bit.

01:37 So if i divide by two, then that's going to be x minus 4, y minus 4, and then half of 224 is 112.

01:54 Next, i'm going to take this expression for x and substitute it in here.

01:59 So that would be 216 over y minus 4 times y minus 4 is equal to 112.

02:18 Now i'm going to do two things at once.

02:20 I notice that these both have a common factor of 4.

02:25 And i also notice that i have a fraction of y, but nothing else is fractions.

02:29 So i really need to get rid of the fractions, and i'd also like to get rid of the common factor that's in there, just like i got rid of the common two here.

02:37 So to get rid of a common factor, you need to multiply by a fraction, so one -fourth.

02:43 And then to get rid of a y on the bottom, you multiply by y on the top.

02:48 So what we're going to do is on the left side, this will distribute into both terms, and then on the right side it will multiply by here, and the four will divide the 112.

03:04 So four goes into 216, 50, 4.

03:09 Times and the ys cancel out.

03:12 The next term, the four goes into four one times, that's negative one and then times y.

03:18 So negative y.

03:20 This is another factor, so it's not affected by distributing into one factor.

03:29 On this side, we have 112 y over four.

03:34 So i'm going to divide 112 by four.

03:37 4 goes into 11, 2 times.

03:39 That leaves 3, so that's 32, and 4 goes into that 8 times.

03:46 Okay, so this term was divided by 4 and multiplied by y, so we're left with the 28y instead of 112.

03:54 Now we're going to distribute each of the numbers.

03:58 We're going to start with number times number.

04:00 The last time i did this, i forgot the negative, so that's going to be really important that when you do this, you keep the negative, because that changes everything.

04:08 Well, we just divided 216 by 4 to get this.

04:13 So if we're going to multiply by 4, we're going to get 216 again.

04:18 And then i'll do the y terms next.

04:20 So 54 times y, 54 y, and negative y times negative 4, that's positive y.

04:32 And then negative y times negative y, that's negative y squared equals 28 y...

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