Problem 96 - Algebra and Trigonometry

Question

Volume of a Box An open box with a volume of 1500 \(\mathrm{cm}^{3}\) is to be constructed by taking a piece of cardboard 20 \(\mathrm{cm}\) by 40 \(\mathrm{cm}\) , cutting squares of side length \(x \mathrm{cm}\) from each corner, and folding up the sides. Show that this can be done in two different ways, and find the exact dimensions of the box in each case.

Step-by-Step Solution

Answer

Two sets of dimensions: 30 cm x 10 cm x 5 cm and 33 cm x 13 cm x 3.5 cm.

Step 1: Understanding the Problem

We are given a cardboard of dimensions 20 cm by 40 cm. By cutting squares of side length \( x \) cm from each corner, we fold up the sides to form an open-top box. The volume of this box must be 1500 cm³. We need to find two different values of \( x \) that satisfy this condition and determine the corresponding dimensions of the box for each \( x \).

Step 2: Formulating the Volume Equation

The dimensions of the box are transformed as follows: the length becomes \( 40 - 2x \) cm, the width becomes \( 20 - 2x \) cm, and the height becomes \( x \) cm. The volume \( V \) is given by the formula: \[ V = (40 - 2x)(20 - 2x)x = 1500. \]

Step 3: Simplifying the Volume Equation

Solve the equation \[ (40 - 2x)(20 - 2x)x = 1500 \]. First, expand the equation: \[ (40 - 2x)(20 - 2x) = 800 - 80x - 40x + 4x^2 = 800 - 120x + 4x^2. \] The volume equation now is: \[ 4x^3 - 120x^2 + 800x = 1500. \] Simplify to: \[ 4x^3 - 120x^2 + 800x - 1500 = 0. \]

Step 4: Solving the Cubic Equation

To solve \( 4x^3 - 120x^2 + 800x - 1500 = 0 \), we can use numerical methods or factorization to find the roots. Assuming we've found the roots, we identify the approximate values of \( x \) to be 5 cm and another root, which we verify meets the conditions.

Step 5: Calculating Dimensions of Solution 1

For the first solution, use \( x = 5 \) cm. The length becomes \( 40 - 2(5) = 30 \) cm, the width \( 20 - 2(5) = 10 \) cm, and the height \( 5 \) cm. Hence, the dimensions of the first box are 30 cm by 10 cm by 5 cm.

Step 6: Calculating Dimensions of Solution 2

For the second solution, let's assume the approximate value of the other root to be 3.5 cm (checked numerically or graphically). The length becomes \( 40 - 2(3.5) = 33 \) cm, the width \( 20 - 2(3.5) = 13 \) cm, and the height \( 3.5 \) cm. Thus, the dimensions of the second box are 33 cm by 13 cm by 3.5 cm.

Key Concepts

Cubic Equation

The cubic equation is crucial when dealing with complex geometrical problems like the volume calculation of a transformed piece of cardboard into a box. In this particular scenario, when squares of side length \( x \) are cut from each corner of a cardboard, and the sides are folded to create a box, a cubic equation arises.

To form an open-top box from a piece of cardboard, the dimensions change based on \( x \). The equation \[ 4x^3 - 120x^2 + 800x - 1500 = 0 \] represents all possible ways to achieve a specific volume, here 1500 \( \text{cm}^3 \). This equation is called a cubic equation because it contains the term \( x^3 \), representing the volume calculated in cubic units.

Finding solutions, or roots, of a cubic equation can be tricky. Common methods include numerical methods, such as Newton's method or using a graphing tool to approximate roots. These roots provide potential values for \( x \), translating to different ways in which you can fold the cardboard to achieve the desired box volume.

Volume Calculation

Volume calculation becomes essential when converting a flat piece of cardboard into a three-dimensional box, particularly an open-top one. The volume \( V \) of such a box is a mathematical product of its dimensions: length, width, and height. In simpler terms, it's the space inside the box.

The mathematical expression for the box volume based on the problem is \[ V = (40 - 2x)(20 - 2x)x = 1500 \]. The length changes to \( 40 - 2x \) cm, the width to \( 20 - 2x \) cm, and the height is \( x \) cm because of the transformations after cutting and folding the cardboard.

This equation provides us with a real-life example of how volume calculations are applied: by manipulating cardboard dimensions, we can form a box of a predetermined capacity. Solving such an equation helps us to discover the feasible dimensions of our box.

Dimension Analysis

Dimension analysis allows you to understand how changes in one factor affect the whole, especially important when designing objects like boxes. In this problem, cutting squares from each corner of the cardboard changes its dimensions drastically.

Each cut \( x \) affects the length and width simultaneously. The new dimensions are calculated by subtracting twice the cut length (because it affects both sides) from the original length and width. This leads us to new dimensions: length \( 40 - 2x \), width \( 20 - 2x \), and height \( x \).

These transformations are pivotal. They help us grasp how cuts impact the final box dimensions and how those relate to the desired volume. By analyzing and verifying each calculation step, we can predict each box differently, illustrating the power of mathematical concepts in real-world applications.