Ch. 1 - Equations and Inequalities

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

Lial 13th Edition

Ch. 1 - Equations and Inequalities

Problem 27

Chapter 2, Problem 27

Volume of a Box. A rectangular piece of metal is 10 in. longer than it is wide. Squares with sides 2 in. long are cut from the four corners, and the flaps are folded upward to form an open box. If the volume of the box is 835 in.³, what were the original dimensions of the piece of metal?

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Define the width of the original rectangular piece of metal as \(w\) inches. Then, the length is \(w + 10\) inches since it is 10 inches longer than the width.

After cutting out squares of side 2 inches from each corner and folding the flaps, the new dimensions of the box will be: length = \(w + 10 - 2 \times 2 = w + 6\), width = \(w - 2 \times 2 = w - 4\), and height = 2 inches (the side length of the squares cut out).

Write the volume formula for the box using these new dimensions: \(V = \text{length} \times \text{width} \times \text{height} = (w + 6)(w - 4)(2)\).

Set the volume equal to 832 cubic inches and form the equation: \((w + 6)(w - 4)(2) = 832\).

Simplify the equation and solve the resulting quadratic equation for \(w\) to find the width, then use \(w + 10\) to find the length.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Formulating Algebraic Expressions from Word Problems

This involves translating the given real-world scenario into algebraic expressions. Here, defining variables for the width and length of the metal piece and expressing the length as 'width + 10' helps set up equations representing the dimensions after cutting and folding.

Volume of a Rectangular Box

The volume of a rectangular box is found by multiplying its length, width, and height. After cutting squares from the corners and folding, the new dimensions change, so understanding how to adjust length, width, and height based on the cuts is essential to form the volume equation.

Solving Quadratic Equations

Setting up the volume equation leads to a quadratic equation in terms of one variable. Solving this quadratic using factoring, completing the square, or the quadratic formula is necessary to find the original dimensions of the metal piece.

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Volume of a Box. A rectangular piece of metal is 10 in. longer than it is wide. Squares with sides 2 in. long are cut from the four corners, and the flaps are folded upward to form an open box. If the volume of the box is 835 in.3, what were the original dimensions of the piece of metal?

Verified video answer for a similar problem:

Key Concepts

Formulating Algebraic Expressions from Word Problems

Volume of a Rectangular Box

Solving Quadratic Equations

Volume of a Box. A rectangular piece of metal is 10 in. longer than it is wide. Squares with sides 2 in. long are cut from the four corners, and the flaps are folded upward to form an open box. If the volume of the box is 835 in.³, what were the original dimensions of the piece of metal?