Problem 152 - Algebra and Trigonometry

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 length and width of the open box.

Step-by-Step Solution

Answer

The length and width of the open box, after subtracting 6 from the value of \( x \), would be the solution for this problem.

Step 1: Understand the Shape of the Box

The box is an open box, meaning it doesn't have a top. After folding up the sides, the corner squares will form the height, and the left-over rectangle is the base of the box. Looking at the base, the 3 inch squares that are cut out will reduce both the length and the width by twice the value of the cut out square, that is, 2*3=6 inches. Let's denote the side of the original metal sheet as \( x \) inches. Therefore, length and width of the box are \( x-6 \) inches each.

Step 2: Formulate the Volume of the Box

The volume \( V \) of a rectangular box (without a top) can be given by the formula \( V = \) length \( \times \) width \( \times \) height = \((x-6) \times (x-6) \times 3\).The volume of the box is given as 75 cubic inches.

Step 3: Solve for the Length and Width

Now, we set the volume equal to 75 cubic inches: \((x-6) \times (x-6) \times 3 = 75\). Solving this equation will give the value of \( x \), which is the original side of the metal sheet. Once we get the value of \( x \), the length and width can be calculated by subtracting 6 from the value of \( x \).

Step 4: Solve the Equation

To solve for \( x \), we can start by simplifying the equation \((x-6)^2 \times 3 = 75\), which simplifies to \(3x^2-36x+108=75\), and further simplifies to \(3x^2-36x+33=0\). This is a quadratic equation and can be solved for \( x \) either by factoring, completing the square, or using the quadratic formula. After solving the equation, only consider the positive value as distance cannot be negative.