Sherlyn's Garden Area Calculation A Mathematical Problem Solving

Sherlyn's garden presents an intriguing mathematical puzzle. She has a small rectangular garden plot, and we know its perimeter is 24 feet. The relationship between the width and the length adds another layer to the problem: the width is exactly one-half of the length. Our mission is to calculate the area of Sherlyn's garden in square feet. This seemingly simple problem allows us to delve into the world of geometry, particularly the properties of rectangles and their perimeters and areas. We will explore how to translate word problems into mathematical equations, solve those equations, and finally, apply the solutions to find the area. The options provided are A. 8, B. 12, C. 32, and D. 72. Let's embark on this mathematical journey and uncover the correct answer.

Understanding the Problem

To solve this problem effectively, we need to break it down into smaller, manageable parts. The first key piece of information is that the garden is rectangular. This tells us that the opposite sides are equal in length and all angles are right angles (90 degrees). The second crucial piece is the perimeter, which is the total distance around the garden, and we know it's 24 feet. The relationship between the width and length is also vital: the width is half the length. This is a key constraint that will help us define the dimensions of the garden. Finally, we need to find the area, which is the space enclosed within the garden, measured in square feet. Understanding these elements is crucial for setting up the mathematical framework to solve the problem.

Setting up the Equations

Now that we understand the problem, let's translate the information into mathematical equations. This is a critical step in solving word problems. Let's use variables to represent the unknowns. Let 'l' represent the length of the garden and 'w' represent the width. We know that the width is half the length, so we can write this as an equation: w = (1/2)l. We also know the formula for the perimeter of a rectangle: P = 2l + 2w, where P is the perimeter. We are given that the perimeter is 24 feet, so we can write the equation: 24 = 2l + 2w. Now we have two equations with two variables, which we can solve simultaneously. These equations are the foundation for finding the dimensions of Sherlyn's garden. Setting up the equations correctly is paramount to finding the correct solution.

Solving for Length and Width

With our equations in place, we can now solve for the length and width of the garden. We have two equations: w = (1/2)l and 24 = 2l + 2w. A common method for solving simultaneous equations is substitution. Since we know w in terms of l, we can substitute (1/2)l for w in the second equation. This gives us 24 = 2l + 2(1/2)l, which simplifies to 24 = 2l + l. Combining like terms, we get 24 = 3l. To solve for l, we divide both sides of the equation by 3, resulting in l = 8. So, the length of the garden is 8 feet. Now that we know the length, we can find the width using the equation w = (1/2)l. Substituting l = 8, we get w = (1/2)(8), which simplifies to w = 4. Therefore, the width of the garden is 4 feet. We have successfully determined the dimensions of Sherlyn's garden: 8 feet long and 4 feet wide.

Calculating the Area

Now that we know the length and width of the garden, we can easily calculate its area. The area of a rectangle is given by the formula A = l * w, where A is the area, l is the length, and w is the width. We found that the length (l) is 8 feet and the width (w) is 4 feet. Substituting these values into the formula, we get A = 8 * 4, which simplifies to A = 32. Therefore, the area of Sherlyn's garden is 32 square feet. This calculation is the final step in solving the problem, and it directly answers the question posed. We have successfully used our knowledge of geometry and algebra to find the area of the rectangular garden.

Conclusion

In conclusion, we have successfully determined the area of Sherlyn's garden by carefully analyzing the given information, setting up appropriate equations, and solving them systematically. We started by understanding the problem, identifying the key elements such as the rectangular shape, the perimeter, and the relationship between the width and length. We then translated this information into mathematical equations, using variables to represent the unknowns. By using the method of substitution, we solved for the length and width of the garden, finding them to be 8 feet and 4 feet, respectively. Finally, we applied the formula for the area of a rectangle (A = l * w) to calculate the area, which turned out to be 32 square feet. Therefore, the correct answer is C. 32. This problem demonstrates the power of mathematical reasoning and problem-solving skills in real-world scenarios. By breaking down complex problems into smaller, manageable steps, we can arrive at the correct solution. This exercise highlights the importance of understanding geometric concepts, algebraic techniques, and the ability to apply them effectively.

Therefore, the final answer is C. 32