Question

Suppose you are the manager of a sheet metal shop. A customer asks you to manufacture 10,000 boxes, each box being open on top. The boxes are required to have a square base and a 9-cubic-foot capacity. You construct the boxes by cutting out a square from each corner of a square piece of sheet metal and folding along the edges. (a) Find the dimensions of the square to be cut if the area of the square piece of sheet metal is 100 square feet. (b) Could you make the box using a smaller piece of sheet metal? Make a list of the dimensions of the box for various pieces of sheet metal.

Short answer

The side length of the base is approximately 1.62 feet and the height is approximately 3.44 feet.

Step-by-step solution

- Understand the Problem

The goal is to find the dimensions of the square cut from the corners of a piece of sheet metal to create an open-top box with a square base and a volume of 9 cubic feet. We also need to verify if it's possible with a smaller piece of sheet metal.

- Set Up the Equations

Let the side length of the square base be denoted as \( x \) and the height of the box as \( h \). The volume of the box (V) is given by \( V = x^2 \times h \). Since the volume is 9 cubic feet, we have: \[ x^2 \times h = 9 \]

- Express Height in Terms of Base

From the volume equation \( x^2 \times h = 9 \), solve for \( h \): \[ h = \frac{9}{x^2} \]

- Consider the Area of Sheet Metal

The area of the original square piece of sheet metal is 100 square feet. Let the side length of this sheet be \( y \). Hence, \( y^2 = 100 \) or \( y = 10 \) feet. The square to be cut from each corner will have the dimensions \( y - 2h \).

- Relate the Sheet Size to Cuts

Given the box height formula \( h = \frac{9}{x^2} \) and the size of the sheet \( y = 10 \), substitute \( h \) into \( y = x + 2h \): \[ 10 = x + 2 \left( \frac{9}{x^2} \right) \]

- Solve for \( x \)

Multiply through by \( x^2 \) to clear the fraction: \[ 10x^2 = x^3 + 18 \] Re-arrange to form a cubic equation: \[ x^3 - 10x^2 + 18 = 0 \]

- Solve the Cubic Equation

Use numerical methods or a graphing calculator to find the appropriate root for the cubic equation. The root is approximately \( x \approx 1.62 \). Therefore, \( h \approx \frac{9}{1.62^2} \approx 3.44 \).

- Verify Smaller Sheets

To verify if smaller sheets can be used, calculate dimensions for various values less than 10 feet. For each \( y' < 10 \), solve for possible box dimensions.

Key concepts

Volume Calculation

In this exercise, we need to create an open-top box with a square base that must have a certain volume. The volume of a box is calculated using the formula: \[ V = \text{base area} \times \text{height} \] For a square base, the area is given by \[ x^2 \] where \ x \ is the length of the side of the square base. The volume requirement is \ 9 \text{ cubic feet} \ (from the problem statement). Thus, the equation becomes: \[ x^2 \times h = 9 \] This equation will guide us to find the dimensions of the square base and the height.

Algebraic Equations

Using algebraic manipulations, we can express one element in terms of others. From the volume equation, we can isolate the height \ h \: \[ h = \frac{9}{x^2} \] After establishing this relationship, we need to consider the area of the sheet metal from which we are cutting the box. Given an area of \ 100 \text{ square feet} \, set the side length \ y \: \[ y^2 = 100 \] So, \ y = 10 \text{ feet} \. The length of each side of the square base plus \ 2h \ (because \ h \ is added on each side when folding) must equal \ y \. Therefore, our equation becomes: \[ y = x + 2h = 10 \] Substitute \ h = \frac{9}{x^2} \ in the equation to form: \[ 10 = x + 2 \times \frac{9}{x^2} \] This equation involves both \ x \ and \ x^2 \ and is solved by clearing fractions. Multiply by \ x^2 \: \[ 10x^2 = x^3 + 18 \] Leading to the cubic equation: \[ x^3 - 10x^2 + 18 = 0 \]. Solving this cubic equation using numerical methods or a graphing calculator gives \ x \ approximately \ 1.62 \text{ feet} \

Geometry Applications

Geometry helps us understand the shape and dimensions of objects. In this exercise, geometrical principles inform us how to fold the sheet metal into a box. When cutting squares from the corners of a square sheet (turning edges into height), you transform the material into a box. Based on our earlier steps: \[ h = \frac{9}{1.62^2} \] This height means we'll cut squares of \ 3.44 \text{ feet} \ height to form our open-top box. Checking smaller sheets, the process repeats with smaller \ y \ values. Ensuring all dimensions must still satisfy the volume requirement, any smaller \ y' \ will need corresponding adjustments to the \ x + 2h \ relation maintaining the cubic equation approach.