Problem 70 - Algebra for College Students

Question

Solve each problem. Cereal box. An independent marketing research agency has determined that the best box for breakfast cereal has a height that is 6 inches (in.) larger than its thickness and a width that is 5 in. larger than its thickness. If such a box is to have a volume of 112 in. \(^{3}\), then what should the thickness be?

Step-by-Step Solution

Answer

The thickness should be 2 inches.

Step 1: Identify given variables

Let the thickness of the box be denoted as x (in.). The height of the box is x + 6 (in.), and the width is x + 5 (in.).

Step 2: Write the volume equation

The volume of a rectangular box is given by the product of its dimensions: height, width, and thickness. Therefore, the volume V (in cubic inches) can be expressed as:\[ V = (x + 6)(x + 5)x \]

Step 3: Substitute the given volume

We know the volume of the box is 112 cubic inches. Substitute this value into the equation:\[ 112 = (x + 6)(x + 5)x \]

Step 4: Expand and simplify the equation

First, expand the expression on the right side:\[ 112 = (x + 6)(x + 5)x \]\[ 112 = (x^2 + 5x + 6x + 30)x \]\[ 112 = (x^2 + 11x + 30)x \]

Step 5: Set up the cubic equation

Simplify the equation further by multiplying through by x:\[ 112 = x^3 + 11x^2 + 30x \]

Step 6: Rearrange to set the equation to zero

Move 112 to the left side to set the equation to zero:\[ x^3 + 11x^2 + 30x - 112 = 0 \]

Step 7: Solve the cubic equation

Solve the cubic equation by finding the roots. One trial method is to use the Rational Root Theorem or synthetic division. Checking fractions, we find:\[ x = 2 \] is a root.

Step 8: Verify the solution

Substitute x = 2 back into the original dimensions to double-check the volume calculation. We have height = 8 in., width = 7 in., thickness = 2 in.\[ 8 \times 7 \times 2 = 112 \]

Key Concepts

Volume of a Rectangular Prism

Understanding the volume of a rectangular prism is crucial for solving this problem. A rectangular prism, or a box shape, has three dimensions: height, width, and thickness. The volume of such a prism is found by multiplying these three dimensions together.
The formula for volume, V, is given by:
\( V = \text{height} \times \text{width} \times \text{thickness} \)
In this exercise, the height is 6 inches more than the thickness, and the width is 5 inches more than the thickness. Using these expressions allows us to set up and solve a cubic equation.

Cubic Equation

A cubic equation is a polynomial equation of degree three. It has the general form:
\( ax^{3} + bx^{2} + cx + d = 0 \)
Where a, b, c, and d are constants, and the highest exponent of x is 3. Solving a cubic equation means finding the value(s) of x that make the equation true.
In the exercise, the cubic equation was derived by expressing the volume in terms of x (thickness) and setting it equal to 112 cubic inches:
\( x^{3} + 11x^{2} + 30x - 112 = 0 \)
This equation can be solved using various methods such as factoring, graphing, the Rational Root Theorem, or synthetic division.

Rational Root Theorem

The Rational Root Theorem helps in finding potential rational solutions (roots) of a polynomial equation. According to the theorem, any rational solution, expressed as a fraction \( \frac{p}{q} \), must have 'p' as a factor of the constant term (last term) and 'q' as a factor of the leading coefficient (first term).
For the equation \( x^{3} + 11x^{2} + 30x - 112 = 0 \):
* Constant term (d): -112 (p must be a factor of -112)
* Leading coefficient (a): 1 (q must be a factor of 1)
This means potential rational roots can be \( \pm 1, \pm 2, \pm 4, \pm 8, \pm 16, \pm 28, \pm 56, \pm 112 \). These roots are tested to see which, if any, satisfy the equation.

Synthetic Division

Synthetic division is a shortcut method for dividing polynomials, especially useful for checking whether a certain value is a root of a polynomial equation.
Steps to perform synthetic division:
1. Write down the coefficients of the polynomial.
2. Choose a candidate root (from the Rational Root Theorem).
3. Perform synthetic division and observe the remainder.
If the remainder is zero, the candidate is a root.
Using synthetic division for our cubic equation \( x^{3} + 11x^{2} + 30x - 112 = 0 \) with the candidate root \( x = 2 \), it is found that 2 is indeed a root:
The division yields a remainder of zero, confirming that \( x = 2 \) is a solution.
This solution verifies the correct dimensions: height = 8 in., width = 7 in., and thickness = 2 in.