Problem 32 - Basic Technical Mathematics with Calculus

Question

Solve the given problems. Use a calculator to solve if necessary. In finding one of the dimensions \(d\) (in in.) of the support columns of a building, the equation \(3 d^{3}+5 d^{2}-400 d-18,000=0\) is found. What is this dimension?

Step-by-Step Solution

Answer

The dimension \(d\) is 20 inches.

Step 1: Identify the Equation

We start with the cubic equation given in the problem, which is \( 3d^3 + 5d^2 - 400d - 18000 = 0 \). This equation is in terms of \( d \), which represents one of the dimensions of the columns in inches.

Step 2: Examine Possible Factors

To find the real roots of the equation, we will look for integer roots that can be calculated directly by testing possible factors of the constant term \(-18000\). Possible factors include integers that divide \(-18000\).

Step 3: Use Rational Root Theorem

The Rational Root Theorem suggests that the possible rational roots of this equation are the factors of \(-18000\) (constant term), divided by the factors of \(3\) (leading coefficient). Test these potential roots to find a valid solution.

Step 4: Test Possible Roots

Test some of the likely integer factors; let's try: \(d = 20\). Substitute \(d = 20\) into the expression to check if it satisfies the equation:\[3(20)^3 + 5(20)^2 - 400(20) - 18000 = 0\]Calculate: \[3(8000) + 5(400) - 8000 - 18000 = 0 \24000 + 2000 - 8000 - 18000 = 0 \0 = 0\]Since this holds true, \(d = 20\) is indeed a root of the polynomial.

Step 5: Confirm the Solution

Given calculations and testing, \(d = 20\) satisfies the polynomial equation. Therefore, it is the valid dimension of the support column that fulfills the given equation.

Key Concepts

Polynomial Roots

Polynomial roots are solutions to polynomial equations. In a polynomial expression like \(3d^3 + 5d^2 - 400d - 18000 = 0\), roots refer to the values of \(d\) that make the equation equal zero. These roots can be real or complex numbers. For a cubic equation, which is a type of polynomial equation with a degree of three, there are always exactly three roots, counting multiplicities, but not all of them might be real.

Identifying the roots of a polynomial is crucial in solving the equation. Here, our task was to find the root that represents the dimension \(d\), which is one of the key measurements of the support columns in the building. By determining the polynomial roots, we found the value of \(d = 20\), which satisfies the given equation. Understanding polynomial roots allows us to evaluate and solve real-life problems like this one with calculations or through estimation.

Rational Root Theorem

The Rational Root Theorem is a handy tool used to predict possible rational roots of a polynomial equation. According to the theorem, any rational solution, or root, of a polynomial is a fraction in the form \( \frac{p}{q} \), where \(p\) is a factor of the constant term and \(q\) is a factor of the leading coefficient.

Applying this to our cubic polynomial equation, the constant term is \(-18000\) and the leading coefficient is \(3\). The possible rational roots are thus the factors of \(-18000\) divided by the factors of \(3\). This gives us a structured list of potential candidates to test, significantly narrowing down the search field.

• Factors of \(-18000\) include numbers like \(1, -1, 2, -2, 3, -3, 4, -4, 5,...,\) and so on, up to 18000 itself.
• Factors of \(3\) are \(1, -1, 3, -3\).

Testing these possible rational roots with the initial polynomial equation allows us to pinpoint the correct one, as we did when we found \(d = 20\).

Solving Equations

Solving equations, especially polynomial equations like \(3d^3 + 5d^2 - 400d - 18000 = 0\), involves finding all possible values that \(d\) can take. There are different methods we can use such as:

• Factoring: Taking out common factors to simplify the equation, although not applicable here.
• Synthetic Division: Once a root is found, division helps simplify the higher-degree polynomial to a quadratric one, making it easier to find remaining roots.
• Using the Rational Root Theorem: As applied here to test integer candidates.
• Utilizing graphing calculators or computers: Helpful for visualizing roots or confirming solutions when analytical methods are complex.

In our exercise, we used the Rational Root Theorem and tested values methodically by substituting them into the polynomial, ultimately finding \(d = 20\), which solved the equation effectively.

Mathematics Education

Mathematics education often focuses on equipping students with the ability to solve real-world problems using mathematical methods. Understanding concepts like polynomial roots, the Rational Root Theorem, and solving equations forms the foundation of algebra, which appears throughout various fields in both pure and applied mathematics.

Encouraging practice with different polynomial problems helps students develop problem-solving skills and logical reasoning. It also familiarizes them with tools that are helpful in calculus and beyond. Educators use such exercises to demonstrate not just calculation techniques, but also to foster critical thinking and analytical skills.

• **Engagement with practical problems**: Solving equations like our cubic one helps see math’s relevance in engineering or architectural contexts.
• **Building blocks for advanced mathematics**: Understanding polynomial functions and roots is key for more complex topics in higher education.

Incorporating these foundational concepts in mathematics education ensures that students recognize the value and application of algebra in diverse areas.