Problem 75 - Algebra 1

Question

Match the problem with the formula needed to solve the problem. Then use the Guess, Check, and Revise strategy or another problem-solving strategy to solve the problem. Area of a rectangle \(\quad A=l w \quad\) Distance \(\quad d=r t\) Simple interest \(\quad I=P r t \quad\) Volume of a cube Temperature \(\quad C=\frac{5}{9}(F-32)\) Surface area of a cube \(S=6 s^{2}\) You want to construct a patio of 80 square feet with a length of 10 feet. What is the width of the patio?

Step-by-Step Solution

Answer

The width of the patio is 8 feet.

Step 1: Identify the correct formula

Looking at the provided formulas, the task asks for the width of a patio, indicating that it implies a rectangle shape. Hence, the formula to use is that for the area of a rectangle, which is \( A=l w \). The goal is to find the width \( w \). The area \( A \) and the length \( l \) are given in the problem.

Step 2: Substituting the given values

Substitute the given values into the formula: 80 = 10 * \( w \).

Step 3: Solve for Width

Solving the equation for \( w \), yields \( w = \frac{80}{10} = 8 \) feet. Thus, the width of the patio is 8 feet.

Key Concepts

area of a rectangle

To solve the problem of finding the width of a patio, it is essential to understand the concept of the area of a rectangle. A rectangle is a shape with four sides and four right angles, usually featuring different lengths for each pair of opposite sides. The formula to calculate the area of a rectangle is given by \( A = l \cdot w \). In this formula, \( A \) stands for the area, \( l \) represents the length, and \( w \) denotes the width.

The area is a representation of the total surface covered by the rectangle, commonly measured in square units such as square feet or square meters. In this case, knowing the area and the length allows us to solve for the width, which is the goal here.

This formula is particularly useful when dealing with real-life problems, such as planning spaces or constructing areas like patios. The simplicity of multiplying length by width makes this one of the most basic yet critical concepts in mathematics. Understanding it helps in visualizing how space is utilized effectively.

algebraic equations

An algebraic equation is a statement that shows the equality between two expressions. In our exercise, we used an algebraic equation to find the width of the patio by substituting the known values of area and length into the rectangle area formula.

In this scenario, the equation formed was: \[ 80 = 10 \times w \] This equation shows that the area, equal to 80 square feet, results from multiplying the length, 10 feet, by the unknown width, \( w \).

Solving algebraic equations involves finding the value of the unknown variable. Here, we solve for \( w \) by dividing both sides of the equation by 10, which isolates \( w \) on one side. Thus, obtaining: \[ w = \frac{80}{10} = 8 \]
Understanding how to manipulate and solve these equations is essential in algebra. It allows you to determine unknown values from known quantities. Many problems in math, science, and even everyday life scenarios can be broken down into algebraic equations and solved systematically using the right techniques.

step-by-step solutions

Using a step-by-step solution is a powerful problem-solving strategy that helps in organizing thoughts and methods for arriving at a correct answer. It breaks down complicated problems into manageable steps, making them easier to understand. Here's how it was applied in our problem:

• Step 1: Identify the correct formula - Start by examining the problem to determine which formula fits best. Knowing you need the width of a rectangle's area involves using \( A = l \cdot w \).


• Step 2: Substitute known values - Insert the given values into the identified formula. Substitute the area (80) and length (10) into \( A = l \cdot w \) to get \( 80 = 10 \cdot w \).


• Step 3: Solve for the unknown - The final step is to solve the equation for \( w \). By dividing 80 by 10, we find that \( w = 8 \) feet.

Each of these steps helps focus on particular aspects of the problem, preventing oversight, and a logical flow assists in clearer comprehension. Additionally, if errors occur, it's easier to retrace steps and identify mistakes. Thus, employing step-by-step solutions not only aids in solving problems but also builds a systematic approach in learning and understanding new concepts.