Sherlyn's Garden Area Calculation Solving A Rectangle Perimeter Problem

In this article, we will embark on a mathematical journey to determine the area of Sherlyn's garden plot. This problem involves understanding the relationship between the perimeter, length, and width of a rectangle, and then applying this knowledge to calculate the area. We will break down the problem step-by-step, providing a clear and concise solution. This exploration will not only solve the specific problem but also enhance your understanding of geometric concepts and problem-solving strategies. So, let's dive in and unravel the dimensions and area of Sherlyn's garden.

H2: Understanding the Problem: Perimeter and Dimensions

The problem states that Sherlyn has a small rectangular garden plot with a perimeter of 24 feet. The key piece of information here is that the width of the garden is exactly $\frac{1}{2}$ of its length. This relationship between width and length is crucial for solving the problem. To begin, let's define the variables we will use:

• Let 'l' represent the length of the garden in feet.
• Let 'w' represent the width of the garden in feet.

Given that the width is half the length, we can express the width in terms of the length as:

• w = $\frac{1}{2}$l

The perimeter of a rectangle is the total distance around its sides, which is calculated by the formula:

• Perimeter = 2l + 2w

We know the perimeter is 24 feet, so we can set up the equation:

• 24 = 2l + 2w

Now, we can substitute the expression for 'w' ($\frac{1}{2}$l) into the perimeter equation. This substitution will allow us to express the equation in terms of a single variable, 'l', which we can then solve. The process of substitution is a fundamental technique in algebra and is widely used in solving various mathematical problems. By replacing 'w' with its equivalent expression, we simplify the equation and bring ourselves closer to finding the dimensions of Sherlyn's garden. This step is essential for unlocking the solution and demonstrating the power of algebraic manipulation.

H2: Solving for Length and Width: A Step-by-Step Approach

Now, let's substitute w = $\frac{1}{2}$l into the perimeter equation:

• 24 = 2l + 2($\frac{1}{2}$l)

Simplify the equation:

• 24 = 2l + l
• 24 = 3l

Now, divide both sides by 3 to solve for 'l':

• l = 24 / 3
• l = 8 feet

So, the length of the garden is 8 feet. Now that we have the length, we can find the width using the relationship w = $\frac{1}{2}$l:

• w = $\frac{1}{2}$ * 8
• w = 4 feet

Thus, the width of the garden is 4 feet. We have successfully determined both the length and the width of Sherlyn's garden plot. This process involved using the given information about the perimeter and the relationship between the length and width to set up and solve an algebraic equation. The ability to manipulate equations and solve for unknowns is a crucial skill in mathematics and has applications in various real-world scenarios. By breaking down the problem into smaller steps, we made the solution more accessible and easier to understand. Now that we have the dimensions, we can move on to the final step: calculating the area.

H2: Calculating the Area: The Final Step

Now that we know the length (l = 8 feet) and the width (w = 4 feet) of Sherlyn's garden, we can calculate the area. The area of a rectangle is given by the formula:

• Area = length * width

Substitute the values we found:

• Area = 8 feet * 4 feet
• Area = 32 square feet

Therefore, the area of Sherlyn's garden is 32 square feet. This final calculation brings us to the solution of the problem. We have successfully used the given information and the principles of geometry to determine the area of the garden plot. The area represents the total surface enclosed within the boundaries of the rectangle and is a fundamental concept in both mathematics and practical applications, such as gardening, construction, and design. Understanding how to calculate area is essential for solving a wide range of problems and making informed decisions in various contexts. With this calculation, we have completed our exploration of Sherlyn's garden and demonstrated the power of mathematical reasoning.

H2: Conclusion: Sherlyn's Garden Area Revealed

In conclusion, by carefully analyzing the given information and applying the principles of geometry, we have determined that the area of Sherlyn's garden is 32 square feet. This problem highlights the importance of understanding the relationships between perimeter, length, width, and area in rectangles. We used algebraic techniques to solve for the unknown dimensions and then calculated the area using the appropriate formula. This exercise not only provides the answer to the specific question but also reinforces the fundamental concepts of mathematics and problem-solving. The ability to break down complex problems into smaller, manageable steps and apply the correct formulas is a valuable skill that can be used in various aspects of life. Sherlyn's garden problem serves as a practical example of how mathematical concepts can be applied to real-world scenarios, making learning more engaging and meaningful. We hope this exploration has clarified the process and enhanced your understanding of geometric calculations.

The correct answer is C. 32.