Problem 113 - College Algebra

Question

A machine produces open boxes using square sheets of metal. The figure illustrates that the machine cuts equal-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 length of the side of the open square-bottom box.

Step-by-Step Solution

Answer

The length of the side of the open square-bottom box is approximately 13.10 inches.

Step 1: Translating the Problem into Mathematical Terms

Let the side length of the initial square sheet be x (inches). After cutting the corners, the resulting open box will have width and length (x-4) inches, and height 2 inches (the side length of the removed squares). The volume, \(V\), of such a box can be represented by the formula \(V = \text{length} \times \text{width} \times \text{height}\) in cubic inches.

Step 2: Setting Up the Equation

Plug in the values for length, width, and height into the volume formula to get an equation representing the volume of the box, \(200 = (x - 4) \times (x - 4) \times 2\). This can be simplified to \(200 = 2x^2 -16x + 16\).

Step 3: Solving for the Side Length

First, divide the entire equation by 2 to simplify the equation: \(100 = x^2 - 8x + 8\). Then, rearrange the equation to form a quadratic equation: \(x^2 - 8x + 8 - 100 = 0\). Simplifying the equation further, we get \(x^2 - 8x - 92 = 0\). Now, use the quadratic formula \(x = \frac{-b\pm\sqrt{b^2 - 4ac}}{2a}\) to solve for the value of x where a=1, b=-8, and c=-92.

Step 4: Determining the Suitable Value for x

The quadratic formula provides two solutions. However, since the side length cannot be negative, we select the positive value as the length of the side of the open square-bottom box.

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool used to find the roots of a quadratic equation, which is an equation that can be written in the form of \(ax^2 + bx + c = 0\). The formula itself is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a\), \(b\), and \(c\) are coefficients from the quadratic equation, and the \(\pm\) symbol indicates that there are generally two solutions to the equation - one for the plus, and one for the minus.

When applying the quadratic formula, it's important to accurately determine the values for \(a\), \(b\), and \(c\), and then carefully perform each mathematical operation step by step. After calculating the discriminant, \(b^2 - 4ac\), students can find out whether there are two real solutions, one real solution, or no real solutions depending on whether the discriminant is positive, zero, or negative, respectively. In the context of the example problem, using the quadratic formula simplifies the process of solving for the side length of the square metal sheet needed to construct the box with a given volume.

Volume of a Box

The volume of a box is found by multiplying its length, width, and height. When the sides are turned up to form the box, these measurements become important for calculating how much space is inside. For example, when metal sheets are used to create boxes, as in the exercise, knowing the volume is key to ensuring that the box will fulfill its intended purpose.

To calculate the volume of an open box with square base cut from a larger square sheet of material, after cutting out the squares at each corner, the sides become the height, and the new length and width are both the initial side length minus twice the height. Here, the formula simplifies to \(V = (x - 2h)^2h\), where \(x\) is the original side length of the square piece of material, and \(h\) is the height of the box after the corners are cut out. The value of the volume dictates the resulting dimensions of the box that can be produced from the material.

Quadratic Equation Applications

Quadratic equations are not just an abstract mathematical concept; they have numerous practical applications in various fields such as physics, engineering, finance, and even in everyday problem-solving situations. For instance, they can be used to model trajectories, optimize profits, and measure forces. In the case of the open box from the exercise, the quadratic equation is applied to find the right size of the metal sheet that will result in the desired volume of the box.

Students can use quadratic equations to evaluate real-life scenarios where a certain outcome is desired, such as finding the dimensions that maximize the volume of a packaging box while minimizing the material used. Quadratic equations often come into play in optimization problems where one needs to maximize or minimize values under certain constraints, making them very relevant and useful in planning and decision-making processes.

College Algebra

College algebra is a course that typically covers a variety of key mathematical concepts, including the study of polynomials, functions, systems of equations, and inequalities. It serves as a critical foundation for many higher-level mathematics courses and applied fields. The quadratic formula, for example, is a primary tool in this subject area and is essential for solving quadratic equations, one of the main topics in college algebra.

Understanding how to manipulate algebraic expressions and solve for unknowns is essential not only in math but also in other scientific and engineering disciplines. The exercise about finding the size of a metal sheet to create a box with a specific volume shows a tangible application of algebra in designing and manufacturing products. Problem-solving skills developed in college algebra are crucial for students in their future careers, as they provide the ability to analyze and solve complex problems systematically.