Problem 371 - Precalculus

Question

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

The height of the box is \( 5x^2 + 12x - 4 \).

Step 1: Recall the Volume Formula

The volume of a box can be calculated using the formula: \( \text{Volume} = \text{length} \times \text{width} \times \text{height} \). We need to find the height and we are given the volume, length, and width.

Step 2: Substitute Known Values into the Formula

Substitute the known values into the volume formula: \( 10x^3 + 30x^2 - 8x - 24 = 2 \times (x+3) \times \text{height} \). We are now looking to solve this equation for the height.

Step 3: Simplify the Equation

First, multiply the length and width: \( 2 \times (x+3) = 2x + 6 \). Substitute this back into the equation for height calculation: \( 10x^3 + 30x^2 - 8x - 24 = (2x + 6) \times \text{height} \).

Step 4: Divide the Polynomial

To solve for height, divide the polynomial \(10x^3 + 30x^2 - 8x - 24\) by \(2x + 6\). Perform polynomial long division for this step.

Step 5: Polynomial Division Result

After performing the polynomial division of \( 10x^3 + 30x^2 - 8x - 24 \) by \( 2x + 6 \), the quotient is \( 5x^2 + 12x - 4 \). Thus, this simplifies to the height of the box.

Step 6: Express the Height Algebrically

The height of the box can be expressed algebraically as \( 5x^2 + 12x - 4 \). This is the result of the division from the previous step.

Key Concepts

Volume of a Box

The volume of a box is a concept that helps determine how much space is inside the box. When you know the dimensions of the length, width, and height, you can calculate the volume using the formula:
\[\text{Volume} = \text{length} \times \text{width} \times \text{height}.\]This formula expresses the relationship between the three dimensions and the volume inside the box. It shows how changing any single dimension affects the overall space the box can hold.

• Length: The longest side of the box.
• Width: The distance of the shorter side at the base.
• Height: How tall the box is.

If, in any problem, you're required to find one of these dimensions (like the height in our example), you can rearrange the formula if given the volume and the other two dimensions. This requires applying algebraic techniques to solve for the unknown value.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operators (like addition or subtraction) that represent a value. These expressions follow algebraic rules and can represent real-world scenarios such as calculating dimensions in geometry. In our exercise, the volume of the box is presented as an algebraic expression:
\[10x^3 + 30x^2 - 8x - 24.\]This expression is known as a polynomial because it consists of powers of a variable (in this case, \(x\)) combined using addition and subtraction. Each separate term in a polynomial has its own degree, determined by the variable's power.

• Constant: A number without a variable, here it is \(-24\).
• Linear term: A term with the variable raised to the power of one, like \(-8x\).
• Quadratic term: A term with the variable squared, such as \(30x^2\).
• Cubic term: A term with the variable cubed, like \(10x^3\).

Algebraic expressions are key in setting up and solving equations related to real-world problems like measuring space or understanding growth patterns.

Factoring Polynomials

Factoring polynomials is a technique of simplifying expressions by breaking them down into smaller, more manageable parts (called factors). It's like reversing multiplication. In our problem, we factored a cubic polynomial to find the height of the box.
When you are given a polynomial and told to find one factor (like the height), you can often perform polynomial division to isolate the unknown factor. Here’s how polynomial division works:

• Identify the divisor, here it's the result of multiplying length and width, \(2x + 6\).
• Divide the volume polynomial, \(10x^3 + 30x^2 - 8x - 24\), by this divisor.
• The quotient is the simpler polynomial that represents the factor we’re interested in. In this case, the quotient is \(5x^2 + 12x - 4\), representing the height.

Polynomial division can seem challenging at first, but with practice, it becomes easier to manage. Recognize that it is a useful tool not just for theoretical math problems but also for practical applications, like calculating the dimensions of a box when working with algebraic expressions.