Problem 70 - 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 \(10 x^{3}+30 x^{2}-8 x-24,\) length is 2 width is \(x+3 .\)

Step-by-Step Solution

Answer

Express the height as \( H = \frac{5x^2 - 8}{x+3} \).

Step 1: Understand the Volume Formula

The volume of a rectangular box can be expressed as the product of its length, width, and height. Algebraically, this is given by the equation: \[ V = L \times W \times H \] where \( V \) is the volume, \( L \) is the length, \( W \) is the width, and \( H \) is the height.

Step 2: Write the Volume Expression

We know the volume of the box is given by \( 10x^3 + 30x^2 - 8x - 24 \). According to the problem, the length (\( L \)) is 2 and the width (\( W \)) is \( x + 3 \). Therefore, we can write the equation: \[ 10x^3 + 30x^2 - 8x - 24 = 2(x + 3)H \] where \( H \) is the height we need to express algebraically.

Step 3: Divide the Volume by Length and Width

To isolate \( H \), we need to divide the volume by the product of the length and the width. Therefore, the height \( H \) can be expressed as: \[ H = \frac{10x^3 + 30x^2 - 8x - 24}{2(x + 3)} \] This equation for \( H \) is derived by dividing both sides of the volume equation by \( 2(x + 3) \).

Step 4: Simplify the Expression

Simplify \( \frac{10x^3 + 30x^2 - 8x - 24}{2(x + 3)} \) through polynomial division. Divide the polynomial \( 10x^3 + 30x^2 - 8x - 24 \) by \( 2(x + 3) \) using long division or synthetic division to find \( H \). After completing the division, you'll find \( H = \frac{5x^2 - 8}{x+3} \).

Key Concepts

Volume of a Box

The volume of a box is an important concept in geometry and algebra. It refers to the amount of space inside a three-dimensional object. For a rectangular box, the volume is determined by multiplying its length, width, and height together. Mathematically, this relationship is expressed as:\[ V = L \times W \times H \]Where:

• \( V \) is the volume
• \( L \) is the length
• \( W \) is the width
• \( H \) is the height

This formula allows you to calculate the volume if you know any three of the four elements. This is useful for a variety of problems, from designing shipping containers to solving algebraic expressions.

Polynomial Division

Polynomial division is a method used to divide polynomials similar to how you divide integers. In algebra, it can be essential when trying to simplify expressions or solve equations. In our exercise, polynomial division is used to find the height of a box by dividing the given polynomial volume by the known length and width. The steps for polynomial division:

• Identify the dividend (the polynomial to be divided) and the divisor (the polynomial you're dividing by).
• Divide the first term of the dividend by the first term of the divisor to get the first term of the quotient.
• Multiply the entire divisor by this new quotient term.
• Subtract this result from the dividend.
• Repeat the process with the new polynomial formed until no terms remain.

This process provides a clear way to break down and simplify a complex polynomial into smaller, more manageable parts. Specifically, when you divide the volume polynomial by the product of the length and width, you can find expressions for unknown dimensions, like the height.

Rectangular Box Dimensions

Understanding the dimensions of a rectangular box is crucial for solving many algebraic problems. In this exercise, the dimensions - length, width, and height - are used to express the relationships in the volume formula.Given:

• The length of the box is 2.
• The width of the box is \( x + 3 \).

You need to find the height. By substituting these values into the volume formula as shown earlier, you form an equation. To solve for the unknown height, knowing the other dimensions lets you isolate and express the height algebraically.In everyday contexts, understanding box dimensions helps in practical tasks such as packing, construction, or even decorating a space. Algebraically, it's an exercise in manipulating and solving expressions, enhancing problem-solving skills in mathematics.