A box with no top is to be constructed from a piece of...

Name: Problem 63, Chapter 6: Intermediate Algebra
Uploaded: 2020-07-25T18:11:52-08:00
Duration: 3 min 20 s
Channel: Heather Zimmers
Description: A box with no top is to be constructed from a piece of cardboard whose length measures 6 in. more than its width. The box is to be formed by cutting squares that measure 2 in. on each side from the four corners and then folding up the sides. If the volume of the box will be 110 in.', what are the dimensions of the piece of cardboard? (FIGURE CANNOT COPY)

A box with no top is to be constructed from a piece of cardboard whose length measures 6 in. more than its width. The box is to be formed by cutting squares that measure 2 in. on each side from the four corners and then folding up the sides. If the volume of the box will be 110 in.', what are the dimensions of the piece of cardboard? (FIGURE CANNOT COPY)

Key Concepts

Solving Polynomial Equations

After formulating the algebraic equation based on geometric constraints, the next step is to solve it. Such equations, often polynomial in nature, require methods like distribution, factoring, or applying formulas to find the unknown dimensions that satisfy the conditions of the problem.

Algebraic Equation Formulation

This key concept involves translating the geometric relationships and constraints into an algebraic equation. By expressing the dimensions of the box in terms of the original piece of material and the modifications made (like cutting out specific shapes), one can create an equation that represents the desired properties, such as a specific volume.

Geometric Modeling in Construction

This concept involves visualizing and representing a three-dimensional object using a two-dimensional material. In problems like constructing a box from a flat piece of cardboard, it requires understanding how modifications such as cutting squares from the corners and folding influence the final shape and dimensions of the box.

Volume Calculation

This concept focuses on determining the space within a three-dimensional shape. When constructing an open-top box, the volume is calculated by multiplying the altered length and width of the base by the height, which is typically determined by the size of the cut-out squares. The volume calculation links physical dimensions to the overall capacity of the box.

Transcript

00:02 Okay, so we're starting with a rectangular sheet of cardboard and six or two inch squares are cut from the corners.

00:12 And then it's folded up to make an open top box.

00:15 So the open top box would look something like this and its height would be two.

00:24 And then this dimension right here is represented by this and this dimension here is represented by this.

00:34 Okay, so the green part would be w -mine, minus 4 because the entire thing was w and we have two here and we have two here.

00:43 So w minus 4 is one of the dimensions of the box.

00:50 And then for the red part, the red part would be take the entire side, w plus 6, subtract 2, subtract 2, and you get w plus 2.

01:03 So now we know all the dimensions of the box and we know the volume of the box is 110 cubic inches.

01:09 And to get the volume of a box, you take the length times the width times the height.

01:22 And we know the length and the width and the height.

01:24 And it doesn't matter what order we write these in.

01:27 We have w plus 2 times w minus 4 times 2.

01:32 Okay, that's going to equal 110.

01:37 Let's divide both sides of the equation by 2 just to make the numbers easier to work with.

01:42 And we have w plus 2 times w minus 4 equals 55.

01:47 Okay, we'll solve this equation for w...

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