A rectangular piece of cardboard, whose area is 216 square...

Name: Problem 85, Chapter 8: College Algebra
Uploaded: 2021-05-06T18:35:33-08:00
Duration: 4 min 10 s
Channel: Gregory Higby
Description: A rectangular piece of cardboard, whose area is 216 square centimeters, is made into an open box by cutting a 2 -centimeter square from each corner and turning up the sides. See the figure. If the box is to have a volume of 224 cubic centimeters, what size cardboard should you start with?

A rectangular piece of cardboard, whose area is 216 square centimeters, is made into an open box by cutting a 2 -centimeter square from each corner and turning up the sides. See the figure. If the box is to have a volume of 224 cubic centimeters, what size cardboard should you start with?

Key Concepts

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.

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.

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.

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 I just need to think a little bit about this problem, so you have a cardboard, and we know, i'll just call this, i don't know, length times a width.

00:12 So i know length times width and the problem is equal to 216.

00:18 They tell us that in the problem.

00:21 And we cut a two centimeter square from each side.

00:27 Now think about it, you have to cut off two centimeters on each side, so when you make the three -dimensional box when you fold things up hopefully you can see what i'm trying to draw is this side length is actually l minus 4 because there's two on each side two over here two over here and the same thing with w it's going to be w minus 4 and then when you start folding this up it's going to have a height of 2 so when you do length times width times height um for the volume um to be 224, we have an equation like that.

01:14 Now the way i would do this problem might be a little bit differently than somebody else, but i would start by dividing the two over to get rid of that.

01:22 Just make that number a little bit easier, so that's 112.

01:25 Now on this side, i could multiply l times w minus 4l minus 4w plus 16.

01:40 And what i might actually do is just replace a l times w with what it's equal to, this 216.

01:46 So we have 216 plus another 216.

01:52 What would that mean? it's like 206 plus 16 would be 232 minus 4l minus 4w will be equal to 112 still.

02:08 I get a calculator out because i'm afraid of me to do their own.

02:11 It should be 120 as i subtract over.

02:17 120 is equal to adding 4l and 4w over and 120 divided by 4 gives me 30...