Question

A cistern is in the shape of a hollow cube whose inside dimension is \(s\) and whose outside dimension is \(S .\) If it is made of concrete of density \(d\), its weight is \(d S^{3}-d s^{3} .\) Factor completely.

Short answer

The weight of the cistern factored completely is \(d(S-s)(S^2+sS+s^2)\).

Step-by-step solution

Identify Common Factors

Observe that the term for weight is a difference of two cubes: \(d S^{3}-d s^{3}\). Both terms have a common factor of \(d\).

Factor Out the Common Factor

Factor out the \(d\) from both terms. This gives us \(d(S^{3}-s^{3})\).

Apply the Difference of Cubes Formula

Recognize that \(S^{3}-s^{3}\) is the difference of cubes and can be factored by the formula \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\). Apply this to factor \(S^{3}-s^{3}\) as \(S-s)(S^2+sS+s^2)\).

Combine Factored Terms

Combine the extracted \(d\) factor with the factored difference of cubes to get the final expression \(d(S-s)(S^2+sS+s^2)\).

Key concepts

Technical Mathematics

Technical Mathematics involves applying mathematical techniques to solve problems related to engineering, physics, and other applied sciences. In the context of factoring the difference of cubes, technical mathematics equips students with strategies to simplify complex expressions, which are often encountered in scientific and engineering calculations.

For example, determining the weight of a hollow concrete cistern involves not only geometric considerations but also an understanding of algebraic manipulation. By identifying common factors and applying specific factoring formulas, students can derive expressions that may be more useful for further calculations or analysis. This problem-solving approach is essential in technical fields where precision and efficiency are paramount.

Polynomial Factoring

Polynomial factoring is a cornerstone of algebra that deals with breaking down polynomial expressions into the product of simpler polynomials. This technique is foundational in simplifying expressions and solving equations. Factoring the difference of cubes is a specific instance where polynomial factoring is used.

The process involves recognizing patterns in the polynomial, like the difference of cubes, and then applying a formula to simplify it. Understanding how to identify common factors, as demonstrated in the weight computation of the cistern, allows students to factor efficiently. Mastery of these techniques is crucial, as it facilitates problem-solving across various areas of mathematics and applied sciences.

Cube of Binomials

The cube of binomials refers to an expression of the form \(a^3 + b^3\) or \(a^3 - b^3\), where \(a\) and \(b\) are binomials. This concept is closely related to the difference of cubes, which can be factored using a specific formula. For \(a^3 - b^3\), the formula is \(a - b)(a^2 + ab + b^2)\), as seen in our example with the dimensions \(S\) and \(s\) of the cistern.

Understanding the cube of binomials provides a deeper grasp of the structure within polynomials. It enables students to recognize these patterns and apply the correct factoring technique, simplifying complex algebraic expressions and allowing for straightforward interpretation and solution of related problems.

Density in Mathematics

In mathematics, density typically refers to a measure of how much of a particular quantity (like mass) is within a given volume. In the case of the cistern, the concept is applied to calculate the weight based on the density of concrete. In algebra, multiplying the density \(d\) by the volume difference \(S^3 - s^3\) gives us the difference in mass between the larger and smaller cubes representing the cistern's outer and inner volumes, respectively.

This context of density in mathematics is about applying algebraic methods to solve real-world problems. By factoring expressions that include density, students learn the practical applications of polynomial factoring and how mathematical concepts can be used to get insights about physical attributes such as weight.