Problem 80 - Introductory Algebra for College Students

Question

A machine produces open boxes using square sheets of metal. The machine cuts equal-sized squares measuring 3 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 75 cubic inches, find the size of the length and width of the open box.

Step-by-Step Solution

Answer

The length and width of the open box will each be 5 inches.

Step 1: Equation Formulation

First, we must understand the equation to solve this problem. The volume \(V\) of a box is given by \(V = lwh\), where \(l\) and \(w\) are its length and width, and \(h\) is its height. The length and the width of the box will be equal, Let's denote them by \(L\). The length of the cut square is given by 3 inches, which will be the \(h\). We can substitute \(L\) and \(h = 3\) into the formula for the volume.

Step 2: Setting Up the Equation

Substitute \(w = w = L\) and \(h = 3\) into the volume equation \(V = lwh\). This will result in \(V = L * L * 3\), which simplifies to \(V = 3L^2\).

Step 3: Solving the Equation

We know from the problem statement that the volume of the box is 75 cubic inches. Thus, we can plug this into our equation \(V = 3L^2\), obtaining the following equation to solve: \(75 = 3L^2\). To isolate \(L\), divide both sides of the equation by 3 to get \(L^2 = 25\). The solution to this equation is \(L = 5\) since length cannot be negative.

Step 4: Finding the Side Lengths

This gives us that the length \(L\) is equal to 5 inches. Since it's a square, the width will also be 5 inches.

Key Concepts

Algebraic Equations

Algebraic equations are powerful tools in mathematics that allow us to represent relationships between different quantities. In the context of our problem, we are interested in finding the dimensions of an open box. The key equation we use is for volume, which is usually given as \( V = lwh \), where \( l \) is the length, \( w \) is the width, and \( h \) is the height.

In this specific case, since the open box has a volume of 75 cubic inches, and the height from the cuts is fixed at 3 inches, we simplify the equation to \( V = 3L^2 \) because the length and width are the same and denoted by \( L \). This equation represents a quadratic relationship which is crucial for solving the problem.

Solving this equation helps find the side lengths required to meet the volume requirement, illustrating how algebraic equations help us translate geometric problems into solvable mathematical expressions.

Problem Solving

Problem solving involves identifying the relevant details and using logical reasoning to find a solution. For the problem at hand, we have to understand both the physical and mathematical aspects.

Here's the approach to tackle this problem:

• First, identify the formula needed: Here, the volume formula \( V = lwh \).
• Recognize the constraints: The cut squares dictate the height of the box, fixed at 3 inches.
• Translate these into equations: Use the given volume (75 cubic inches) to establish a relationship with the unknown side \( L \).

By substituting into the volume equation, \( V = 3L^2 \), we simplify the problem to solving a quadratic equation.

Solving \( 3L^2 = 75 \) leads us to find \( L = 5 \) through steps of algebraic manipulation. This structured approach demonstrates effective problem solving skills.

Open Box

An open box is a box without a lid, which is often formed by cutting squares from each corner of a flat sheet and folding up the sides. In this problem, the open boxes are produced by removing squares of side 3 inches from a square piece of metal.

This means the height of the box will be precisely 3 inches after forming the box from the sheet. The essential feature of an open box in this situation is understanding that it has no top, which influences its volume calculation.

The process of forming the box involves determining the size of the original sheet needed to ensure the final box has the required volume, taking into account the reduction in length and width due to the cuts. This makes the open box concept crucial as it determines the dimensions through the length and width being adjusted while establishing the specific volume requirement based on the real, physical constraints given by the cutting process.