Problem 67 - 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 \(12 x^{3}+20 x^{2}-21 x-36\) length is \(2 x+3,\) width is \(3 x-4\)

Step-by-Step Solution

Answer

Perform polynomial division of \(12x^3 + 20x^2 - 21x - 36\) by \((2x+3)(3x-4)\) to express the height algebraically.

Step 1: Recall the formula for the volume of a box

The volume of a box (rectangular prism) is calculated as the product of its length, width, and height. Thus, the formula to find the volume is \( V = \text{length} \times \text{width} \times \text{height} \). Therefore, the height can be isolated and expressed as \( \text{height} = \frac{V}{\text{length} \times \text{width}} \).

Step 2: Set up the equation with given values

Plug in the given values into our formula for height. The volume \( V \) is \(12x^3 + 20x^2 - 21x - 36\), the length is \(2x + 3\), and the width is \(3x - 4\). Therefore, the equation for the height becomes: \( \text{height} = \frac{12x^3 + 20x^2 - 21x - 36}{(2x + 3)(3x - 4)} \).

Step 3: Simplify the expression

To simplify it further, perform polynomial division of \(12x^3 + 20x^2 - 21x - 36\) by the product of \( (2x + 3)(3x - 4) \). First, multiply the polynomials \((2x+3)(3x-4)\) using distribution (or foil method): - Distribute \(2x\) to \(3x\) and \(-4\): \((2x) \times (3x) = 6x^2\) and \((2x) \times (-4) = -8x\).- Distribute \(3\) to \(3x\) and \(-4\): \(3 \times 3x = 9x\) and \(3 \times (-4) = -12\).Combine these to get \(6x^2 + x - 12\).

Step 4: Polynomial division

Now, divide \(12x^3 + 20x^2 - 21x - 36\) by \(6x^2 + x - 12\). The division gives you a quotient which represents the height of the box.

Step 5: Perform polynomial division

Perform synthetic or long division to find the quotient. In this scenario, we would need to closely follow through with either method ensuring correct coefficients and achievable steps through subtraction align with the polynomial division technique. Eventually, the quotient from division will reflect a simplified polynomial indicating the height.

Step 6: Final Expression for Height

Upon performing the division, the height function is simplified into a polynomial expression in terms of \(x\).

Key Concepts

Volume of a Box

Calculating the volume of a box is essential in understanding how much space a particular box can contain inside it. The volume is calculated by multiplying the length, width, and height of the box.

In mathematical terms, we express this as:

• The formula for volume: \( V = \text{length} \times \text{width} \times \text{height} \)
• To find the height, we can rearrange it: \( \text{height} = \frac{V}{\text{length} \times \text{width}} \)

In our specific problem, the given volume \( V \) is a polynomial: \( 12 x^3 + 20 x^2 - 21 x - 36 \).
The expressions for the length and width are \( 2x+3 \) and \( 3x-4 \) respectively.

So now you see the volume as more than just numbers. It’s an expression that encapsulates the box’s space needs algebraically. To find out the missing dimension, such as height, we must dive into polynomial division along with rational expressions.

Rational Expressions

Rational expressions are the quotient of two polynomials, similar to fractions but in an algebraic form. Understanding these is crucial for operations involving polynomials, like dividing the box's polynomial volume by the product of its given dimensions.

Here's the step-by-step guide:

• You create a rational expression to isolate the height: \( \frac{12x^3 + 20x^2 - 21x - 36}{(2x + 3)(3x - 4)} \).
• Observe that the rational expression denotes the polynomial volume divided by the product of its length and width expressions. This is useful in finding the height algebraically.

When you approach these expressions, think of them as multi-component fractions. Simplifying them can help reveal simpler polynomial forms.

In our scenario, simplifying involves polynomial division, which transforms a challenging expression into a more digestible polynomial reflecting the missing dimension of the box—the height.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operations (like addition and multiplication). They are key building blocks in algebra that you use when solving problems about volumes, areas, and other properties.

To deal with algebraic expressions effectively:

• Recognize each part of the expression, such as coefficients, variables, and their powers.
• Learn how to manipulate these expressions using operations like addition, subtraction, or multiplication.
• In polynomial division, break down complex expressions into easier-to-handle components.

In our example, the volume expression was \( 12x^3 + 20x^2 - 21x - 36 \). This showcases how complex expressions can be restructured by dividing them through polynomial division. As you simplify this, your result is the height expressed as another polynomial.

Understanding and mastering algebraic expressions is essential in translating real-world problems into solvable mathematical equations. It helps simulate and solve problems like finding a box's height when the volume is given as a complex algebraic expression. This approach transforms abstract concepts into tangible solutions.