Solving Rectangle Problems: Perimeter, Area, And Cost

Hey math enthusiasts! Today, we're diving into a classic geometry problem involving a rectangular plot of land. We'll be using our knowledge of perimeter, area, and a little bit of algebra to solve for some key values. Get ready to put on your thinking caps and let's get started!

(I) Finding the Value of x

Let's break down the problem. We're given a rectangular plot of land with a length of (2x + 5) meters and a width of (x - 10) meters. The perimeter of this plot is 80 meters. Our first task is to find the value of x. Remember, the perimeter of a rectangle is the total distance around it, and it's calculated using the formula: Perimeter = 2 * (length + width).

So, let's plug in the given values into the formula: 80 = 2 * ((2x + 5) + (x - 10)). Now, we need to solve this equation for x. This involves a few steps of algebraic manipulation. First, let's simplify the expression inside the parentheses: (2x + 5) + (x - 10) = 3x - 5. Now our equation looks like this: 80 = 2 * (3x - 5). Next, distribute the 2 across the terms inside the parentheses: 80 = 6x - 10. To isolate the term with x, add 10 to both sides of the equation: 80 + 10 = 6x. This simplifies to 90 = 6x. Finally, to solve for x, divide both sides by 6: 90 / 6 = x. Therefore, x = 15. So, we've successfully found the value of x! It's an important step, as it will help us find the dimensions of the rectangle, and hence the area and cost. Always remember to double-check your calculations to ensure accuracy. Small mistakes can easily happen, so it's a good practice to go back and verify your work.

Now that we know the value of x is 15. We can calculate the dimensions of the rectangle. The length of the rectangle is (2x + 5) meters, substituting x = 15, we get (2*15 + 5) = 35 meters. The width of the rectangle is (x - 10) meters, substituting x = 15, we get (15 - 10) = 5 meters. This means that our rectangle has a length of 35 meters and a width of 5 meters. This will be very useful information in the next section where we calculate the area. The key to solving problems like these is to break them down into smaller, manageable steps. By carefully applying formulas and algebraic techniques, we can conquer even the most complex geometric challenges. Keep practicing, and you'll become a pro in no time! Remember to always label your units (meters in this case) to keep things clear and organized. This not only aids in understanding but also helps in avoiding common mistakes. Keep up the great work!

(II) Calculating the Area of the Plot

Alright, now that we know the value of x and we know the length and width of our rectangular plot (35 meters and 5 meters respectively), it's time to find the area. The area of a rectangle is calculated by multiplying its length by its width. The formula is: Area = length * width.

In our case, the length is 35 meters and the width is 5 meters. So, the area of the plot is 35 meters * 5 meters = 175 square meters (m²). Remember, the area is always expressed in square units because it represents the space covered by the two-dimensional shape. This value is critical for determining the cost of weeding the entire plot. Always remember the units, it is really important when calculating areas and volumes, and keeps your calculations accurate. Double-checking your calculation is always a good strategy to keep your answers precise. It's really easy to make mistakes during calculations, therefore double-check your calculations for accuracy. Always write down your steps, in the long run, this will help you to understand and learn the process better.

So, the area of the rectangular plot is 175 square meters. We're making great progress in solving this problem! We've found the value of x and calculated the area of the plot. With this crucial information, we're now ready to tackle the final part of our problem: calculating the cost of weeding. Understanding the relationship between perimeter, length, width, and area is fundamental in solving this type of problem. Practicing these types of problems is very useful. It will enhance your skills and your proficiency in mathematics.

(III) Determining the Cost of Weeding

Here comes the final step! We're given that the cost of weeding is $0.24 per square meter. We've already calculated the area of the plot to be 175 square meters. To find the total cost of weeding, we simply multiply the area by the cost per square meter: Total Cost = Area * Cost per square meter.

So, the total cost is 175 m² * $0.24/m² = $42.00. Therefore, the cost of weeding the entire plot is $42.00. This is how we wrap up the problem! We started with the perimeter, used it to find x, then found the area, and finally calculated the total cost. Isn't it satisfying to see how different mathematical concepts come together to solve a real-world problem?

Keep in mind that this type of problem can be altered in many ways, with different shapes and costs. For instance, the shape could be changed to a triangle, or a circle. The cost could be based on various factors, such as the type of grass or the labor involved. The possibilities are endless. These problems will help strengthen your mathematical thinking skills. As you solve more problems, you will become more confident and accurate. Always remember to break down complex problems into simpler steps.

Conclusion

Congratulations! We've successfully solved the problem involving the rectangular plot of land. We found the value of x, calculated the area, and determined the cost of weeding. This problem demonstrates the practical application of basic geometric concepts and algebraic techniques. The key takeaways from this problem are the formulas for perimeter and area of a rectangle, the ability to solve a linear equation for a single variable, and the importance of paying attention to units. This exercise is great for solidifying your understanding of these concepts. Keep practicing, and you'll become proficient in solving various geometry problems! Remember, practice makes perfect. The more problems you solve, the more confident and skilled you'll become. So, keep up the great work, and don't hesitate to tackle new challenges.

We hope you found this explanation helpful and that it gave you a better understanding of how to solve similar problems. Keep exploring the world of mathematics, and never stop learning!

For more information on the area and perimeter of a rectangle, you can check out this resource: Khan Academy - Area and perimeter