Problem 79 - Introductory Algebra for College Students

Question

A machine produces open boxes using square sheets of metal. The figure illustrates that the machine cuts equa sized squares measuring 2 inches on a side from the corners and then shapes the metal into an open box by turning up the sides. If each box must have a volume of 200 cubic inches, find the size of the length and width of the open box. (THE IMAGES CANNOT COPY)

Step-by-Step Solution

Answer

The size of the length and width of the open box is 10 inches.

Step 1: Understanding the Problem

A sheet of metal is being used to create an open box by removing 2 inch squares from each corner and folding up the sides. Since the squares being cut are the same size, you can infer that the length and width of the box will be the same, making it a square. The height of the box will be the length of the side of the cut squares, which is given as 2 inches. The volume of the box is given as 200 cubic inches.

Step 2: Applying the Volume Formula

The formula for the volume of a box is \(v = lwh\), where l is the length, w is the width, and h is the height. In this case, we know that the width and length are the same, so we can call them 'x'. So the volume v equals to \(x * x * h\) . Hence, the formula becomes, \(v = x^2 * h\). Given that the volume v is 200 cubic inches, and the height h is 2 inches, you can substitute these into the formula and solve for x.

Step 3: Solving for x

Substitute the volume and the height into the equation, to get \(200 = x^2 * 2\). Simplifying gives \(x^2 = 200/2\), which becomes \(x^2 = 100\). The length and width x is the square root of 100, which is 10. So the size of the length and width of the open box is 10 inches.

Step 4: Validating the solution

To confirm this answer, substitute 10 inches for x in the volume calculation: \(v = x^2 * h\), and check that the volume equals 200 cubic inches. So \(v = 10*10*2 = 200\), the volume is indeed 200 cubic inches, confirming that the solution is correct.

Key Concepts

Algebra

Algebra is a branch of mathematics that uses symbols and letters to represent numbers and quantities. In solving the problem of the volume of a box, algebraic skills are crucial. Let's break it down.

You start with the formula for the volume of a box:

• \( V = lwh \), where \( l \) is the length, \( w \) is the width, and \( h \) is the height.

In this problem, both the length and width of the box are equal, because square side sections are cut from each corner of a rectangular sheet, making it a square base box. Let's call this side \( x \).

Substitute the given height \( h = 2 \) inches into the formula, and since the sides are equal, set the formula to \( V = x^2 \, h \). Given the box must have a volume of 200 cubic inches, substitute \( V = 200 \) into the formula:

• \( 200 = x^2 \, \times \, 2 \)
• Simplify to: \( x^2 = 100 \)

To solve for \( x \), take the square root of both sides of the equation: \( x = \sqrt{100} = 10 \) inches.This means the length and width of the open box are each 10 inches.

Geometry

Geometry, another essential branch of math, deals with the properties and relations of points, lines, surfaces, and solids. For this problem, understanding the geometry of the box is key to solving for dimensions.

Imagine taking a flat square sheet of metal and cutting small squares out of each corner. You're left with tabs that you can fold upright, forming an open box. The box's base remains a square, shaped by the remaining metal.

• The small squares have sides of 2 inches, and these form the height of the box after folding.

This leaves the base as the original sheet's dimension reduced by

• 4 inches (2 inches deducted from each side).

For example, if the original side of the metal sheet were \( x + 4 \), the new dimensions would become \( x \), working as both the length and width in our volume calculation.

This geometric manipulation shows the intersection between geometry's physical shapes and algebra's calculative power.

Problem-Solving

At the heart of mathematics is problem-solving, the art of breaking down a complex problem into understandable parts and solving them systematically. Here's how it works for this box-making exercise.

Firstly, make sure to grasp the requirements of the problem. Understand the box's dimensions and how the cutting and folding process affects them. Being able to visualize these changes can significantly aid in understanding.

Next, translate the problem into mathematical equations that reflect these transformations. This is where the volume formula comes in handy. Recognize that the length and width are altered by the removal and folding of the square corners.

• Visual aids like sketches can make it easier to connect the physical object with mathematical calculations.
• Check your work by substantiating results in the original equation and ensuring they satisfy all given conditions.

Finally, validate your solution. In our case, by plugging the solved value for \( x \) back into the equation, we confirm that the volume totals 200 cubic inches, just as the problem requires. Successful problem-solving requires clear thinking and rigorous verification.