Problem 68 - Algebra 1

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 be if you want it to be square?

Step-by-Step Solution

Answer

The side length of the kitchen needs to be approximately 12.25 feet.

Step 1: Formulate the equation

The area of a square is given by the formula \(Area = side^2\). Given that the area of the kitchen is 150 square feet, we can write this formula to represent the kitchen: \(side^2 = 150\). This is the quadratic equation that we need to solve.

Step 2: Solve the quadratic equation

To solve the equation \(side^2 = 150\), we can square root both sides of the equation. The square root of 150 is approximately 12.25. Therefore, \(side = \sqrt{150} \approx 12.25\) feet.

Step 3: Verify the solution

To verify the solution, we substitute \(side = 12.25\) into the original equation. So, \(side^2 = (12.25)^2 = 150\) square feet. This confirms that the side length solution is correct.

Key Concepts

Area Calculation

Calculating the area of a shape is a fundamental concept in geometry. The area represents the space enclosed within the shape's boundaries. For a square, this concept is particularly straightforward because all its sides are equal. The formula to find the area of a square is:

• Area = side × side,
• or simply, Area = side².

In the context of designing floor plans, knowing the desired area assists in determining the possible dimensions of a room. If, for example, a kitchen must have an area of 150 square feet, we use the area formula to ascertain the side length, ensuring that the room will fit the design requirements. Calculating the area helps in resource planning, space efficiency, and understanding how much material is needed for construction. Regularly breaking down these tasks simplistically can build strong foundational knowledge in applications of geometry.

Square Dimensions

The dimensions of a square are defined by its sides, each of which is equal in length. This equality is what sets a square apart from other quadrilaterals. When tasked with designing a room, such as a kitchen as in our example, where we want square dimensions, we start with our known values.

• All sides are equal: side = side = side = side.
• The area, given as a whole, can bring us to knowing individual side lengths.

Determining these dimensions requires breaking down spatial necessities into mathematical equations. The given area of 150 square feet allows us to calculate the side by using the property that defines a square's area: Area = side².
By solving this equation, we can find that each side will be approximately 12.25 feet long, dictating a square room that fulfills the specified area. In architectural planning, understanding geometry and its principles helps design spaces that meet specific criteria efficiently.

Solving Equations

Solving equations is a critical skill that allows us to find unknown values from known quantities. Quadratic equations, such as the one posed by the condition that side² = 150, commonly appear in various mathematical problems. Here’s how we solve it:

• Identify the form of the equation: in this case, side² = 150.
• To isolate the variable, apply the square root to both sides.
• This gives side = √150, which is approximately 12.25.

For this particular equation, taking the square root allows us to solve for the side length of the kitchen.
Once calculated, verifying the result by substituting it back into the original equation ensures the solution is correct. For those learning this process, practicing solving simple equations will build confidence in tackling more complex challenges. Quadratic equations are especially useful as they frequently appear in modeling real-world situations, allowing for precise calculation and prediction related to dimensions, motion, and other phenomena.