Problem 68 - College Algebra

Question

For the following exercises, use the given volume of a box and its length and width to express the height of the box algebraically. Volume is \(18 x^{3}-21 x^{2}-40 x+48\) length is \(3 x-4,\) width is \(3 x-4\)

Step-by-Step Solution

Answer

The height of the box is \(6x^2 + x - 12.\)

Step 1: Understand the Formula for Volume

The volume of a box (rectangular prism) is found using the formula: \( \text{Volume} = \text{Length} \times \text{Width} \times \text{Height}. \) We know both the volume and the expressions for length and width.

Step 2: Set up the Equation

Given the formula for volume, we can express it as: \[ 18x^3 - 21x^2 - 40x + 48 = (3x - 4)(3x - 4) \times \text{Height} \] where \( \text{Height} \) is the unknown expression we need to find.

Step 3: Simplify Length and Width

First, simplify the product of the length and width: \((3x - 4)(3x - 4) = (3x - 4)^2 = 9x^2 - 24x + 16.\)

Step 4: Divide Volume by Product of Length and Width

To find the height, divide both sides of the volume equation by \((3x - 4)^2\): \[ \text{Height} = \frac{18x^3 - 21x^2 - 40x + 48}{9x^2 - 24x + 16}. \] Factor the numerator if possible to ensure it divides evenly.

Step 5: Factor the Numerator

Factor the cubic polynomial \(18x^3 - 21x^2 - 40x + 48\). After factoring, this expression simplifies to \[(3x - 4)(6x^2 + x - 12).\] This shows that the polynomial is divisible by \((3x - 4)\).

Step 6: Simplify to Find the Height

From Step 4's division, \( \text{Height} = \frac{(3x-4)(6x^2 + x - 12)}{(3x - 4)^2}.\) Cancel one \((3x - 4)\) term from both the numerator and the denominator: \( \text{Height} = \frac{6x^2 + x - 12}{3x - 4}. \) But, because \((3x - 4)\) was also a factor in the numerator precisely once, efficiently, the height simplifies to \(6x^2 + x - 12.\)

Key Concepts

Volume of a Box

Understanding the volume of a box is an essential part of geometry and algebra. The volume represents the amount of space inside a three-dimensional object. For a box, also known as a rectangular prism, this is calculated using the formula:
\[ \text{Volume} = \text{Length} \times \text{Width} \times \text{Height}. \]
In this problem, we are given the volume as a polynomial expression, \( 18x^3 - 21x^2 - 40x + 48 \), along with the length and width having the same expression \( 3x - 4 \). The aim is to determine the height as an algebraic expression.
Understanding this formula allows us to rearrange it and solve for any missing dimension, such as height, when the volume and other two dimensions are known. This is where our knowledge of factorization and polynomial division comes into play.

Factorization

Factorization is a method used in algebra to break down larger expressions into simpler components or 'factors' that can be multiplied back together to give the original expression.
For the polynomial \( 18x^3 - 21x^2 - 40x + 48 \), we identified its factors through careful analysis. The given expression was factored as \((3x - 4)(6x^2 + x - 12)\). This step is essential because it helps simplify expressions for further operations like polynomial division.
Here's why factorization is important:

• It Simplifies Complex Problems: By breaking down the polynomial, we make it easier to handle and understand.
• Facilitates Division: Factorization allows us to cancel common factors when dividing polynomials, as seen by eliminating one \((3x - 4)\) during the division.
• Identifies Roots: Factors of a polynomial provide information about its roots, which are points where the polynomial equals zero.

Polynomial Division

Once the polynomial \( 18x^3 - 21x^2 - 40x + 48 \) was factored into \((3x - 4)(6x^2 + x - 12)\), we used polynomial division to determine the height of the box. Polynomial division lets us divide a polynomial by another polynomial, simplifying complex algebraic expressions.
By setting up the division as \( \frac{(3x-4)(6x^2 + x - 12)}{(3x - 4)^2} \), we can simplify it to \( 6x^2 + x - 12 \), since one \((3x - 4)\) cancels out.
The key reasons polynomial division is important are:

• Reduces Complexity: Dividing polynomials reduces complex expressions and provides solutions that we can interpret directly.
• Finds Quotients: It helps in obtaining the quotient when one polynomial is divided by another.
• Essential for Further Calculations: Understanding and performing such division is crucial for solving higher-level algebra problems efficiently.

Mastering polynomial division builds a foundation for more advanced math topics and problem-solving techniques.