Problem 57 - Algebra 1: Concepts and Skills

Question

Make a sketch and write a quadratic equation to model the situation. Then solve the equation. In art class you are designing the floor plan of a house. The kitchen is supposed to have 150 square feet of space. What should the dimensions of the kitchen floor be if you want it to be square?

Step-by-Step Solution

Answer

The dimensions of the kitchen floor should be approximately 12.247 feet by 12.247 feet.

Step 1: Understand the problem and what is being asked.

The problem is asking for the dimensions of a square kitchen floor with an area of 150 square feet.

Step 2: Formulate the mathematical model

Since the kitchen floor is square, it implies that length equals breadth. Let \(x\) denote the dimension of the square-side. Thus, the area of the square, \(A\), is given by \(A = x^2\). The exercise provides that \(A = 150\) square feet. So, the quadratic equation is \(x^2 = 150\).

Step 3: Solve the equation

To find the value of \(x\), take the square root of both sides of the equation: \(x = \sqrt{150}\). The square root of 150 is approximately 12.247. Therefore, each side of the kitchen floor should measure approximately 12.247 feet.

Key Concepts

Solving Quadratic Equations

Understanding how to solve quadratic equations is essential for students as it forms the foundation for various real-life applications. A quadratic equation is an equation of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a \eq 0\). The equation given in the exercise, \(x^2 = 150\), is a simplified quadratic equation with \(a = 1\), \(b = 0\), and \(c = -150\).

To solve such equations, methods like factoring, completing the square, or using the quadratic formula can be employed. In the given problem, since the equation is already in a simple form, taking the square root of both sides is sufficient. It's important to note that when you take the square root of both sides, we consider both positive and negative solutions. However, in real-world scenarios like this exercise, where a dimension is being calculated, the negative solution is not feasible. Therefore, we only use the positive result of the square root to reach our solution.

Area of a Square

The area of a square can be determined by multiplying the length of one side by itself, or \(side^2\). In the context of the exercise, the area of the kitchen is given as 150 square feet. This area value leads us to set up the equation \(x^2 = 150\), where \(x\) is the length of the side of the square-shaped kitchen.

It is crucial to visualize the problem: when we say a floor is square, we imply all the sides are equal. Thus, knowing the area gives us a clear path to find the length of a side. The concept also ties in with the geometric formula for the square's area, reinforcing the link between algebra and geometry. Remember, the units of measurement are also essential, in our case, the result will be in feet, because the area was in square feet.

Mathematical Modeling

Mathematical modeling involves creating mathematical representations of real-world situations to solve problems or predict outcomes. In our exercise, the real-life scenario is the design of a kitchen floor plan. By translating the requirement of a 150 square feet kitchen into a mathematical model, we've created an equation \(x^2 = 150\) to represent the situation.

This process illustrates how mathematical models simplify complex real-world problems into manageable mathematical tasks. The solution to the equation then provides us the practical answer required for our real-life problem - the dimensions of the kitchen. Always aim to understand the situation at hand and devise the equation as a model of the problem, as it will assist in determining what the variables represent and guide us towards the solution.