Calculating The Area Of Shertiar's Rectangular Garden

Understanding the Garden's Dimensions

In this mathematical problem, we are presented with Shertiar's small rectangular garden plot, and our task is to calculate the area of this garden. To do so, we'll need to carefully dissect the information provided and apply some fundamental geometric principles. Let's begin by unraveling the details of this problem. The problem states that the garden has a perimeter of 24 feet. This is a crucial piece of information because the perimeter of a rectangle is directly related to its length and width. Remember, the perimeter is the total distance around the rectangle, which can be calculated by adding up the lengths of all four sides. In a rectangle, opposite sides are equal in length, so the perimeter is essentially twice the length plus twice the width. We're also given another critical clue: the width of the garden is exactly one-half of its length. This relationship between width and length is key to solving the problem. It allows us to express both dimensions in terms of a single variable, which will greatly simplify our calculations. For instance, if we let the length of the garden be represented by the variable 'l', then the width can be represented as 'l/2'. This sets the stage for us to formulate an equation based on the given perimeter. Now, let's translate these pieces of information into mathematical terms. We know that the perimeter (P) of a rectangle is given by the formula: P = 2l + 2w, where 'l' is the length and 'w' is the width. In Shertiar's case, we know that P = 24 feet and w = l/2. Substituting these values into the formula, we get: 24 = 2l + 2(l/2). This equation represents the perimeter of Shertiar's garden in terms of its length. Our next step is to solve this equation for 'l'. Once we find the length, we can easily determine the width using the relationship w = l/2. With both length and width known, we'll be able to calculate the area of the garden, which is the ultimate goal of this problem. The area (A) of a rectangle is simply the product of its length and width: A = l * w. By following these steps, we'll methodically unravel the problem and arrive at the solution for the garden's area.

Setting Up the Equation

To accurately determine the area of Shertiar's rectangular garden plot, we need to set up an equation that incorporates the information provided. As discussed earlier, the perimeter of a rectangle is given by the formula P = 2l + 2w, where 'l' represents the length and 'w' represents the width. In this particular scenario, we know that the perimeter (P) is 24 feet. Moreover, we have the crucial detail that the width (w) is exactly one-half of the length (l). This relationship can be expressed mathematically as w = l/2. Now, we can substitute these values into the perimeter formula to create an equation that involves only one variable, the length 'l'. This substitution is a key step in simplifying the problem and making it solvable. Replacing 'P' with 24 and 'w' with l/2 in the perimeter formula, we obtain the following equation: 24 = 2l + 2(l/2). This equation now encapsulates all the essential information about the garden's dimensions. The left side of the equation represents the total perimeter, while the right side represents the sum of twice the length and twice the width (expressed in terms of the length). The next step is to simplify this equation and solve for 'l'. By doing so, we'll find the length of the garden. Once we have the length, we can easily calculate the width using the relationship w = l/2. With both length and width known, we'll be well-equipped to calculate the area of the garden, which is the product of length and width. The process of setting up the equation is a critical step in problem-solving. It involves translating the given information into mathematical language. A well-formulated equation allows us to apply algebraic techniques to find the unknown quantities. In this case, our equation 24 = 2l + 2(l/2) is the foundation for determining the dimensions of Shertiar's garden and, ultimately, its area. Now that we have the equation, the next phase involves simplifying it and solving for the length 'l'. This will bring us closer to finding the final answer.

Solving for the Length and Width

Having established the equation 24 = 2l + 2(l/2), the next crucial step in determining the garden's dimensions is to solve for the length (l). This involves simplifying the equation and isolating the variable 'l'. Let's begin by simplifying the right side of the equation. We have 2(l/2), which simplifies to just 'l'. This is because multiplying 'l' by 2 and then dividing by 2 effectively cancels out the multiplication. So, our equation now becomes: 24 = 2l + l. Next, we can combine the like terms on the right side. We have 2l + l, which equals 3l. Therefore, our equation simplifies further to: 24 = 3l. Now, to isolate 'l', we need to divide both sides of the equation by 3. This is a fundamental algebraic operation that maintains the equality of the equation while getting 'l' by itself. Dividing both sides by 3, we get: 24/3 = 3l/3, which simplifies to: 8 = l. So, we have found that the length (l) of Shertiar's garden is 8 feet. With the length determined, we can now easily calculate the width (w) using the relationship w = l/2. Substituting l = 8 into this equation, we get: w = 8/2, which simplifies to: w = 4. Therefore, the width of Shertiar's garden is 4 feet. At this point, we have successfully determined both the length and the width of the rectangular garden plot. The length is 8 feet, and the width is 4 feet. These dimensions are essential for calculating the area of the garden, which is the ultimate goal of the problem. The process of solving for the length and width involves applying basic algebraic principles to simplify the equation and isolate the unknown variables. By carefully following the steps of simplification and division, we were able to find the values of 'l' and 'w'. Now that we know the length and width, we are just one step away from calculating the area of the garden.

Calculating the Area

With the length and width of Shertiar's garden successfully determined, we can now proceed to calculate the area of the rectangular plot. As established earlier, the area (A) of a rectangle is given by the formula: A = l * w, where 'l' is the length and 'w' is the width. In our previous calculations, we found that the length (l) of the garden is 8 feet and the width (w) is 4 feet. To find the area, we simply need to multiply these two values together. Substituting l = 8 and w = 4 into the area formula, we get: A = 8 * 4. Performing this multiplication, we find that: A = 32. Therefore, the area of Shertiar's garden is 32 square feet. This is the final answer to the problem. We have successfully calculated the area of the rectangular garden plot by first setting up an equation based on the given perimeter and the relationship between the length and width, then solving for the length and width, and finally applying the area formula. The units of the area are square feet because we are measuring a two-dimensional space. The length and width were given in feet, so their product, the area, is in square feet. The calculation of the area represents the culmination of our problem-solving process. We started with a description of the garden and some key pieces of information, and through a series of logical steps, we arrived at the numerical value of the area. This demonstrates the power of mathematical reasoning and the application of geometric principles to solve real-world problems. In summary, Shertiar's rectangular garden plot has an area of 32 square feet. This is the space available for planting and gardening within the confines of the plot's dimensions.

Final Answer

After carefully working through the problem, we have arrived at the final answer for the area of Shertiar's garden plot. To recap, we were given that the garden has a perimeter of 24 feet and that the width is exactly one-half of the length. Our goal was to determine the area of the garden in square feet. We began by setting up an equation based on the perimeter formula, P = 2l + 2w, and the relationship between the width and length, w = l/2. Substituting the given values into the formula, we obtained the equation 24 = 2l + 2(l/2). Next, we simplified this equation and solved for the length (l). We found that the length of the garden is 8 feet. With the length known, we calculated the width (w) using the relationship w = l/2. This gave us a width of 4 feet. Finally, we applied the area formula, A = l * w, to calculate the area of the garden. Substituting the values we found for length and width, we got A = 8 * 4, which equals 32. Therefore, the area of Shertiar's garden is 32 square feet. This is the numerical answer to the problem, and it represents the amount of space within the garden plot that Shertiar has available for planting and gardening activities. The problem-solving process involved several key steps, including setting up an equation, simplifying the equation, solving for the unknowns (length and width), and applying the appropriate formula to calculate the area. Each step was crucial in arriving at the correct answer. In conclusion, the area of Shertiar's small rectangular garden plot is 32 square feet. This is the solution to the mathematical problem presented, and it demonstrates the application of geometric principles and algebraic techniques to solve real-world scenarios.