Problem 48 - Algebra 2

Question

The Energy Booster Company keeps their stock of Health Aid liquid in a tank that is a rectangular prism. Its sides measure \(x-1\) centimeters, \(x+3\) centimeters, and \(x-2\) centimeters. Suppose they would like to bottle their Health Aid in \(x-3\) containers of the same size. How much liquid in cubic centimeters will remain unbottled?

Step-by-Step Solution

Answer

Unbottled liquid: \( x^3 - 8x + 9 \) cm³.

Step 1: Find the volume of the tank

To find the volume of the tank, use the formula for the volume of a rectangular prism, which is the product of its side lengths: \ Volume of the tank = \( (x-1) \times (x+3) \times (x-2) \).

Step 2: Calculate the volume as a polynomial

Expand the expression \((x-1)(x+3)(x-2)\): 1. Multiply the first two binomials: \ \((x-1)(x+3) = x^2 + 3x - x - 3 = x^2 + 2x - 3\).2. Then multiply this result by the third binomial: \((x^2 + 2x - 3)(x-2)\).3. Perform the multiplication step-by-step: - \(x^2 \times x = x^3\), \(x^2 \times (-2) = -2x^2\) - \(2x \times x = 2x^2\), \(2x \times (-2) = -4x\) - \(-3 \times x = -3x\), \(-3 \times (-2) = 6\)4. Combine like terms: \ \(x^3 - 2x^2 + 2x^2 - 4x - 3x + 6 = x^3 - 7x + 6\). Therefore, the volume of the tank is \(x^3 - 7x + 6\) cubic centimeters.

Step 3: Find the volume of one container

The volume of one container is given as \( 1 \) cubic centimeter (since the problem states that the Health Aid is bottled in \(x-3\) containers of equal size and all bottles together hold the volume).

Step 4: Calculate the number of containers

The number of containers is \( x-3 \). Since each container holds \(1\) cubic centimeter, the total volume they can hold is simply \(x-3\) cubic centimeters.

Step 5: Determine the volume of unbottled liquid

Subtract the total volume of liquid the containers can hold from the volume of the tank: \ \( x^3 - 7x + 6 - (x-3) = x^3 - 7x + 6 - x + 3 \).Combine like terms: \ \( x^3 - 8x + 9 \).Thus, the volume of liquid that will remain unbottled is \( x^3 - 8x + 9 \) cubic centimeters.

Key Concepts

Polynomial Expansion

Polynomial expansion is a method used to expand a product of two or more polynomials into a sum of monomials. In the exercise, we expanded the product of three binomials: \((x-1)\), \((x+3)\), and \((x-2)\). By expanding these, we calculate the total volume of the tank in terms of the variable \(x\).
To start, we first multiply the first two binomials using the distributive property, known as the FOIL method (First, Outside, Inside, Last). This gives us:

• First: \(x \cdot x = x^2\)
• Outside: \(x \cdot 3 = 3x\)
• Inside: \((-1) \cdot x = -x\)
• Last: \((-1) \cdot 3 = -3\)

This simplifies to \(x^2 + 2x - 3\). Next, we multiply this result by the third binomial \((x-2)\).
The multiplication proceeds in a step-by-step manner:

• Multiply each term from the first polynomial by each term of the second polynomial.
• Combine like terms to get the final polynomial expression, \(x^3 - 7x + 6\).

This expanded polynomial represents the volume of the tank.

Binomial Multiplication

Binomial multiplication is a crucial process in algebra, especially in problems requiring polynomial operations, like the one in this exercise. A binomial is an algebraic expression with two terms, typically written as \((a + b)\) or \((a - b)\).
When multiplying binomials, a common technique is to apply the distributive property, which helps in expanding the expression. For the given product \((x-1)(x+3)(x-2)\), we take it step by step:

• Multiplying the first two binomials: \((x-1)(x+3)\).
• Use the distributive property or FOIL method to ensure every term is considered.
• Take the result and multiply it by the third binomial, \((x-2)\).

This meticulous approach guarantees that all terms are accounted for and consequently leads to an accurate expanded polynomial. The resultant expression \(x^3 - 7x + 6\) reflects this precise process and serves as the foundation for further computation in the exercise.

Unbottled Liquid Calculation

The calculation of unbottled liquid requires a thorough understanding of the earlier derived polynomial representing the tank's volume. Once we have the polynomial for the tank's volume, \(x^3 - 7x + 6\), we need to consider the configurations of the containers.
The problem states that the liquid is stored in \(x-3\) containers, each holding a standard volume, in this case, 1 cubic centimeter. Therefore, the total volume the containers could accommodate is exactly \(x-3\) cubic centimeters.
The remaining step is to determine how much liquid stays in the tank after filling all the containers. This involves subtracting the volume held by the containers from the tank's total volume. Here's the formula:
\[\text{Unbottled Volume} = (\text{Volume of the Tank}) - (\text{Volume of all Containers})\]

1. Initially, substitute the values: \(x^3 - 7x + 6 - (x-3)\).
2. Simplify the expression: \(x^3 - 8x + 9\).

Hence, the volume of liquid not poured into the containers is \(x^3 - 8x + 9\) cubic centimeters—this shows the importance of precise polynomial and arithmetic skills in solving real-world problems like inventory management.