Problem 151 - College Algebra Essentials

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 and width of the open box.

Step-by-Step Solution

Answer

The dimensions of the open box are 10 inches in length, 10 inches in width, and 2 inches in height.

Step 1: Define the Variables

Let's denote the side length of the original square sheet of metal is \(x\) inches. After the squares are cut out of each corner, the sides of the box (which will equal the length and width of the base) are also \(x-4\) inches and the height is 2 inches.

Step 2: Write down the Volume formula

The formula for the volume of the box (assuming it is rectangular in shape) is \(V =length \times width \times height\). The question states that the volume of the box should be 200 cubic inches. We substitute these values into the formula: \(200 = (x-4) \times (x-4) \times 2\)

Step 3: Solve the Equation

Solving the equation: \(200 = (x-4)^2 \times 2\). First, divide both sides by 2: \((x-4)^2 = 100\). The next step is to take the square root of both sides to find \(x-4\). This gives us two solutions \(x-4 = 10\) and \(x-4 = -10\). But \(x-4\) cannot be negative because \(x\) represents a length, and length cannot be negative. Hence, we discard the negative solution and solve \(x-4 = 10\) for \(x\), which gives us \(x = 14\).

Step 4: Find the dimensions of the box

Subtract 4 (the cut-out sides) from \(x\) to find the length and width of the box. This gives us dimensions of the open box of \(14-4 = 10\) inches length and width, with a height of 2 inches.

Key Concepts

Volume Formula

Calculating the volume of any 3D object is crucial when designing or measuring for practical applications, like an open box. For a rectangular box, the volume can be calculated using a simple formula:

• The Volume Formula is: \[ V = ext{length} \times ext{width} \times ext{height} \]
• This formula captures how much space is inside the box.

For example, if you know the length, width, and height of a box, you simply multiply these values to find the volume. In our case, the box needed to have a volume of 200 cubic inches, which set the goal for our calculations.
Remember, when dealing with specific designs like open boxes, height is a crucial dimension, affecting how the box can be used or what it can contain.

Rectangular Box

A rectangular box is a common 3D shape with distinct sides. It is important because it has a clear structure: three pairs of parallel sides.

• The basis of a rectangular box's volume stems from its shape.
• Understanding its dimensions helps in determining how inner space is managed.

In open box scenarios, the height becomes a key measurement. With open tops, such boxes can hold different items depending on their size constraints.
Think about everyday objects, like shoeboxes or drawers, all essentially rectangular boxes. Recognizing such forms can aid in grasping calculations related to their structure and functionality.

Solving Equations

Equations often seem daunting, but they are simply statements of balance realized through variables. Solving them is a step-by-step journey to find an unknown value.

• In our situation, we used the equation derived from the volume formula: \[ 200 = (x-4)^2 \times 2 \]
• This was solved by isolating \((x-4)^2\)

First, we simplified the equation by dividing both sides by 2, leaving us with \((x-4)^2 = 100\). Dealing with powers often involves inverse operations, hence taking the square root. This provided us two potential solutions. But only feasible side lengths (positive numbers) are considered in real-world setups.
When solving equations, each action taken maintains the original equation's truth to guide you steadily to the answer.

Length and Width Calculation

Finding the length and width of an open box from the given conditions involves minimal algebra but meticulous understanding.

• We've established that the open box's dimensions have lengths that are reduced from the initial metal sheet.
• Our equation gave us \( x-4 = 10 \).

Given the solved equation and knowing the deductions made from the metal corners (2-inch cuts), finding the new length and width was straightforward. If the total length was originally greater, just subtract the cut dimensions to determine the base.For a practical scenario like this, understanding the theoretical math and how modifications affect dimensions is vital. It ensures accurate measurements and efficient resource usage in industrial settings.