Miguel Rectangular Wood Pieces Solving For Area And Dimensions

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Miguel has several rectangular pieces of wood, all identical in size and shape. He uses 6 of these pieces to form a figure with a perimeter of 80 centimeters. The sum of the base and height of one of the pieces is 12 centimeters. Our goal is to determine the area of one of these rectangular pieces.

Understanding the Problem

This is a geometry problem that combines concepts of perimeter, area, and algebraic relationships. We need to carefully analyze the information given and devise a strategy to find the dimensions of the rectangular piece and then calculate its area. Let's break down the given information:

  • Miguel has identical rectangular pieces of wood.
  • He uses 6 pieces to form a figure.
  • The perimeter of the figure is 80 centimeters.
  • The sum of the base and height of one piece is 12 centimeters.

Our target is to find the area of one rectangular piece. To do this, we need to first determine the dimensions (base and height) of the rectangle.

Setting up Equations

Let's use variables to represent the unknowns. Let:

  • b = the base of one rectangular piece
  • h = the height of one rectangular piece

From the given information, we can set up two equations:

  1. Equation 1 (Sum of base and height): b + h = 12
  2. Equation 2 (Perimeter of the figure): This equation requires a bit more analysis. We need to look at how the 6 pieces are arranged to form the figure. Unfortunately, the problem description does not include an image or a detailed description of the figure's shape. Without knowing the arrangement, it's impossible to directly translate the perimeter into an equation involving b and h. However, we can make some assumptions and explore possible scenarios.

Let's assume, for the sake of explanation, that the figure is arranged in such a way that the perimeter consists of 10 lengths of 'h' and 4 lengths of 'b'. This means that the perimeter equation would look like this:

10h + 4b = 80

This is a hypothetical example, and the actual coefficients of b and h in the perimeter equation would depend on the specific arrangement of the rectangles.

Solving for Base and Height (Hypothetical Example)

Using the hypothetical Equation 2 (10h + 4b = 80), let's demonstrate how we would solve for b and h. We now have a system of two equations with two variables:

  1. b + h = 12
  2. 10h + 4b = 80

We can use various methods to solve this system, such as substitution or elimination. Let's use the substitution method.

From Equation 1, we can express b in terms of h:

b = 12 - h

Now, substitute this expression for b into Equation 2:

10h + 4(12 - h) = 80

Simplify and solve for h:

10h + 48 - 4h = 80

6h = 32

h = 32 / 6 = 16 / 3 (approximately 5.33 cm)

Now, substitute the value of h back into the equation b = 12 - h:

b = 12 - (16 / 3) = (36 - 16) / 3 = 20 / 3 (approximately 6.67 cm)

So, in this hypothetical scenario, the height h would be approximately 5.33 cm and the base b would be approximately 6.67 cm.

Calculating the Area (Hypothetical Example)

Now that we have the base and height (in our hypothetical example), we can calculate the area of one rectangular piece:

Area = b * h

Area = (20 / 3) * (16 / 3) = 320 / 9 (approximately 35.56 square centimeters)

The Importance of the Figure's Arrangement

It's crucial to emphasize that the above calculations are based on a hypothetical arrangement of the rectangular pieces. The actual perimeter equation and, consequently, the final area will depend entirely on how the 6 pieces are arranged to form the figure.

To solve this problem accurately, we would need either a diagram of the figure or a clear description of how the rectangles are connected. For instance, if the rectangles were arranged in a single row, the perimeter equation would be different than if they were arranged in a 2x3 grid. Let's consider another possible arrangement to illustrate this point.

Alternative Arrangement Scenario: Rectangles in a Row

Suppose the 6 rectangles are placed side-by-side in a single row. In this case, the dimensions of the resulting figure would be:

  • Length: 6 times the base of one rectangle (6b)
  • Width: The height of one rectangle (h)

The perimeter of this figure would be:

Perimeter = 2 * (Length + Width) = 2 * (6b + h)

Given that the perimeter is 80 cm, our perimeter equation would be:

2 * (6b + h) = 80

12b + 2h = 80

This is a different Equation 2 than our previous hypothetical example. Let's solve this system of equations:

  1. b + h = 12
  2. 12b + 2h = 80

We can use the substitution method again. From Equation 1, h = 12 - b. Substitute this into Equation 2:

12b + 2(12 - b) = 80

12b + 24 - 2b = 80

10b = 56

b = 5.6 cm

Now, substitute the value of b back into the equation h = 12 - b:

h = 12 - 5.6 = 6.4 cm

In this scenario, the base b is 5.6 cm and the height h is 6.4 cm.

Calculating the Area (Rectangles in a Row)

The area of one rectangular piece would be:

Area = b * h

Area = 5.6 cm * 6.4 cm = 35.84 square centimeters

Notice that the area is different from our previous hypothetical example. This demonstrates how the arrangement of the rectangles significantly impacts the final answer.

General Approach and Key Takeaways

To accurately solve this type of problem, follow these steps:

  1. Understand the Problem: Carefully read and analyze the given information. Identify the unknowns and the relationships between them.
  2. Set up Equations: Use variables to represent the unknowns and translate the given information into mathematical equations. The most critical step is to correctly formulate the perimeter equation based on the figure's arrangement.
  3. Solve the Equations: Use appropriate algebraic techniques (substitution, elimination, etc.) to solve the system of equations and find the values of the unknowns.
  4. Calculate the Area: Once you have the base and height, calculate the area of the rectangle using the formula: Area = b * h.
  5. Consider Different Arrangements: If the problem does not provide a clear description of the figure's arrangement, consider different possibilities and analyze how each arrangement affects the perimeter equation and the final answer.

Conclusion

Miguel's rectangular wood piece problem is a great example of how geometry and algebra can be combined to solve real-world puzzles. The key takeaway is that the arrangement of the pieces is crucial for determining the perimeter and, consequently, the dimensions and area of each piece. Without a clear description or diagram of the figure, it's impossible to provide a definitive answer. However, by understanding the underlying concepts and following a systematic approach, we can analyze different scenarios and arrive at potential solutions.

To accurately determine the area of one rectangular piece in Miguel's problem, we need additional information about the figure's arrangement. A diagram or a detailed description of how the 6 pieces are connected would allow us to formulate the correct perimeter equation and solve for the base and height of the rectangles.

Miguel has several rectangular wooden pieces, all the same size. He forms a figure with a perimeter of 80 centimeters using 6 of these pieces. If the sum of the base and height of one piece is 12 centimeters, what is the area of one of the pieces?