Problem 96 - Elementary Algebra

Question

Yvette wants to put a square swimming pool in the corner of her backyard. She will have a 3 foot deck on the south side of the pool and a 9 foot deck on the west side of the pool. She has a total area of 1080 square feet for the pool and two decks. Solve the equation \((s+3)(s+9)=1080\) for \(s,\) the length of a side of the pool.

Step-by-Step Solution

Answer

The length of a side of the pool is 27 feet.

Step 1: Understanding the Problem

Yvette wants to build a square pool with a 3-foot deck on the south side and a 9-foot deck on the west side. The total area for both the pool and the decks is 1080 square feet. We need to find the side length of the pool, represented by \(s\).

Step 2: Set up the Equation

Given the formula \((s+3)(s+9)=1080\), identify \(s+3\) as the total length, including the pool and the south deck, and \(s+9\) as the total width, including the pool and the west deck.

Step 3: Expand the Equation

Expand the left side of the equation: \((s+3)(s+9) = s^2 + 12s + 27\).

Step 4: Set Up the Quadratic Equation

Equate the expanded form to 1080: \(s^2 + 12s + 27 = 1080\)

Step 5: Subtract 1080 from Both Sides

Move 1080 to the left side to set the equation to zero: \(s^2 + 12s + 27 - 1080 = 0\)

Step 6: Simplify the Equation

Simplify the resulting equation: \(s^2 + 12s - 1053 = 0\)

Step 7: Solve the Quadratic Equation

Solve for \(s\) using the quadratic formula \(s = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Here, \(a = 1\), \(b = 12\), and \(c = -1053\). Plug these values into the quadratic formula: \(s = \frac{-12 \pm \sqrt{144 + 4212}}{2}\).

Step 8: Calculate the Discriminant

Compute the discriminant: \(144 + 4212 = 4356\)

Step 9: Find the Square Root

Take the square root of the discriminant: \(\sqrt{4356}=66\)

Step 10: Find the Solutions

Substitute back into the quadratic formula: \(s = \frac{-12 \pm 66}{2}\). This results in two solutions: \(s = \frac{54}{2} = 27\) and \(s = \frac{-78}{2} = -39\). Since the side length cannot be negative, the solution is \(s = 27\).

Key Concepts

quadratic equations

Quadratic equations are essential in algebra. These equations are written in the form \[ax^2 + bx + c = 0\]. The variable is often represented by \(x\), but in different contexts, like our swimming pool problem, it can be another letter, like \(s\).
The main feature of a quadratic equation is that the highest power of the variable is 2, making it a second-degree polynomial. Quadratic equations appear in many real-world situations.
In our example, Yvette's problem can be modeled with the equation \((s+3)(s+9) = 1080\) because we are calculating the area of a square pool and its decks.

solving equations

Solving quadratic equations can be achieved through several methods. For our swimming pool problem, we follow a series of steps:
First, we transformed the original equation \((s+3)(s+9) = 1080\) by expanding it. After expanding, it becomes \[s^2 + 12s + 27 = 1080\].
The equation then needs to be set to zero by moving 1080 to the left side: \[s^2 + 12s - 1053 = 0\].
Subsequently, we used the quadratic formula, which is \[s = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\], with \(a = 1\), \(b = 12\), and \(c = -1053\). Plugging in these values, we calculate the discriminant and solve for \(s\).
The solutions are \(s = 27\) and \(s = -39\). Since a length cannot be negative, our side length for the pool is \(s = 27\) feet.

geometry and algebra integration

Integrating geometry and algebra allows us to solve practical problems like Yvette's. Here, we used algebra to represent the total area of a pool combined with decks.
The equation \((s+3)(s+9) = 1080\) translates the physical dimensions into a mathematical form.
The terms \(s+3\) and \(s+9\) represent the dimensions of the area, combining the pool's side \(s\) with the additional deck lengths (3 feet and 9 feet).
Algebra helps us manipulate these dimensions and solve the equation, finding the side length of the pool.

discriminant calculation

The discriminant is a vital part of the quadratic formula, shown in \[b^2 - 4ac\]. It enables us to determine the number and type of roots the quadratic equation has.
A positive discriminant indicates two real solutions. In Yvette's problem, the discriminant is \[144 + 4212 = 4356\].
The square root of this discriminant is 66, leading to solutions \(s = \frac{54}{2} = 27\) and \(s = \frac{-78}{2} = -39\).
With a negative side length ruled out, the positive solution gives us \(s = 27\), meaning the pool's side is 27 feet long.