Problem 53 - College Algebra

Question

For the following exercises, consider this scenario: Charlotte has appointed chairperson to lead a city beautification project. The first actis to install statues and fountains in one of the city's parks. The park is a rectangle with an area of \(98 x^{2}+105 x-27 \mathrm{m}^{2},\) as shown in the following figure. The length and width of the park are perfect factors of the area. At the northwest corner of the park, the city is going to install a fountain. The area of the base of the fountain is \(9 x^{2}-25 \mathrm{m}^{2}\) . Factor the area to find the lengths of the sides of the fountain.

Step-by-Step Solution

Answer

The sides of the fountain are \((3x - 5)\) and \((3x + 5)\).

Step 1: Set up the dimensions

The fountain sides are given by the expressions \((3x-5)\) and \((3x+5)\) (or similar expressions from the problem context).

Step 2: Calculate area or perimeter as needed

Apply the appropriate formula using the given polynomial expressions to find the required measurement.

Key Concepts

Difference of Squares

When dealing with the area of the fountain's base, expressed as \(9x^2 - 25\), we notice an application of the difference of squares. The difference of squares is a special mathematical method for factoring expressions, making complicated ones easier to manage. It relates to expressions like \(a^2 - b^2\), which can be simplified to \((a-b)(a+b)\).
This concept is crucial in algebra because it reveals a straightforward way to deal with quadratic expressions involving subtraction between two perfect squares. In our case:

• The expression \(9x^2 - 25\) is recognized as \((3x)^2 - 5^2\),
• which simplifies the equation to \((3x - 5)(3x + 5)\).

This method is particularly useful in solving problems involving areas, as it allows us to break down more complex areas into manageable linear terms, revealing the dimensions necessary for further calculations.

Rectangular Area

Understanding rectangular area is key in solving the given exercise, as it forms the foundation for how we view space in terms of geometry and algebra. A park shaped like a rectangle with an area given by the polynomial \(98x^2 + 105x - 27\) suggests we find its length and width through factoring.
The area of a rectangle is calculated simply as length multiplied by width. When dealing with algebraic expressions as areas, it involves finding the factors of the expression so these factors turn out to represent the dimensions.
In this exercise, you need to factor the polynomial to determine the lengths of the sides. While the polynomial indicates a substantial area, your aim is to simplify it to smaller polynomial factors that reveal the physical dimensions of the rectangle. This highlights the beauty of using algebra to solve real-world spatial problems.

Polynomial Expressions

Polynomial expressions like \(98x^2 + 105x - 27\) and \(9x^2 - 25\), often appear in problems relating to areas and dimensions. By understanding polynomials, you can decipher complex forms into simpler, workable parts.
Polynomials comprise terms containing variables raised to whole number powers and often require careful breaking down through techniques such as factoring. Factoring involves decomposing a polynomial into a product of simpler polynomials.

• For example, in the polynomial \(9x^2 - 25\), recognizing the pattern of difference of squares allows for immediate factoring.
• On a broader scale, accurately factoring polynomial expressions assists in identifying properties like dimensions, roots, or intercepts necessary for problem-solving.

In essence, mastering polynomial expressions empowers you to handle complex algebraic forms by simplifying them to their core, actionable elements across various mathematical contexts, such as calculating areas in geometric problems.