Problem 67 - College Algebra with Corequisite Support

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 \(12 x^{3}+20 x^{2}-21 x-36,\) length is \(2 x+3,\) width is \(3 x-4\)

Step-by-Step Solution

Answer

The height of the box is \(2x + 5\).

Step 1: Understand Volume Formula

The volume of a box is given by the product of its length, width, and height. Mathematically, this is expressed as \( V = l \times w \times h \), where \( V \) is the volume, \( l \) is the length, \( w \) is the width, and \( h \) is the height.

Step 2: Set Up the Equation

Using the volume formula, we can express the height \( h \) as: \( h = \frac{V}{l \times w} \). We know that \( V = 12x^3+20x^2-21x-36 \), \( l = 2x+3 \), and \( w = 3x-4 \). Substitute these values into the equation: \( h = \frac{12x^3 + 20x^2 - 21x - 36}{(2x+3)(3x-4)} \).

Step 3: Polynomial Long Division Setup

To simplify the expression for \( h \), we need to perform polynomial long division. We will divide \( 12x^3+20x^2-21x-36 \) by \((2x+3)(3x-4)\). First, expand \((2x+3)(3x-4)\) to get \(6x^2 + x - 12\).

Step 4: Divide the Polynomials

Perform polynomial long division by dividing \(12x^3 + 20x^2 - 21x - 36\) by \(6x^2 + x - 12\). The first term in the quotient comes from dividing \(12x^3\) by \(6x^2\), which is \(2x\). Multiply \(2x\) by \(6x^2 + x - 12\) and subtract from the original polynomial, then repeat the division for the remaining polynomial.

Step 5: Solve the Division Completely

Continuing with polynomial division after obtaining the first quotient term as \(2x\), the process yields the next term as \(5\). Thus, the quotient simplifies to \(2x + 5\), meaning \( h = 2x + 5 \).

Step 6: Check the Division

Multiply the quotient \(2x + 5\) by \(6x^2 + x - 12\) to ensure it results in the original polynomial \(12x^3+20x^2-21x-36\). The multiplication confirms that we have the correct height.

Key Concepts

Volume of a Box

When calculating the volume of a box, we rely on a simple mathematical relationship: the product of the box's length, width, and height. The volume formula is expressed as \( V = l \times w \times h \). Here, \( V \) stands for volume, \( l \) for length, \( w \) for width, and \( h \) for height. Knowing this formula is crucial when addressing problems involving three-dimensional shapes like boxes, which are essentially rectangular prisms.

In the provided exercise, we are given the volume of the box as a polynomial: \( 12x^3+20x^2-21x-36 \). To determine the height algebraically, we re-arrange the volume formula. This allows us to solve for height \( h \) by dividing the provided volume by the product of the other two given dimensions, length and width. To express this:

• Volume \( V \)
• Length \( l = 2x+3 \)
• Width \( w = 3x-4 \)

This leads us to the equation \( h = \frac{V}{l \times w} \), a key step in understanding the depth of polynomial applications in geometric contexts.

Polynomial Division

Polynomial division is a process similar to long division with numbers, but instead, it involves expressions with variables. It is an important concept in algebra, especially when simplifying expressions as in our exercise.

In our specific scenario, to find the height of the box, we must perform polynomial division on the volume expression \( 12x^3+20x^2-21x-36 \) by the expanded form of the product of length and width, \((2x+3)(3x-4)\). Expanding \((2x+3)(3x-4)\) results in a single trinomial: \(6x^2 + x - 12\), which becomes our divisor.

The division process involves:

• Finding how many times the leading term of the divisor can divide the leading term of the dividend (e.g., \(12x^3\) by \(6x^2\) gives the initial result \(2x\)).
• Multiplying the entire divisor by this result and subtracting it from the original polynomial.
• Repeating the cycle for any remaining polynomial parts until completely simplified.

This systematic approach ensures a step-by-step simplification of polynomials, breaking down complex expressions into manageable components.

Quotient Calculation

The calculation of a quotient in polynomial division reveals significant insights into algebraic expressions. It simplifies complex polynomials into usable forms, much like finding the height of the box problem we are tackling.

As we proceed with division, the result, or quotient, has a significant role. In our case, during the polynomial division of \(12x^3+20x^2-21x-36\) by \(6x^2+x-12\), we discovered that the quotient is \(2x+5\). This quotient is a key value, as it represents the height \( h \) of the box.

Understanding why the quotient is the height involves recognizing that each term in the polynomial represents a dimension of the geometric shape. When you divide the entire polynomial volume by the other two dimensions (length and width), the remaining term(s) directly equate to the third dimension, thus providing the height.

Quotient calculations in algebra help connect real-world dimensions to abstract polynomials, making mathematics applicable in everyday tasks like measuring and constructing objects.