Find The Dimensions That Produce The Maximum Floor Area For A One-story House That Is Rectangular In Shape And Has A Perimeter Of 131 Ft.A. 10.92 Ft × 32.75 Ft 10.92 \, \text{ft} \times 32.75 \, \text{ft} 10.92 Ft × 32.75 Ft B. $3275 , \text{ft} \times 3275 ,

Introduction

When designing a one-story house, one of the primary considerations is the floor area. A larger floor area can provide more space for living, entertainment, and other activities. However, the dimensions of the house are limited by its perimeter, which is the total length of its boundaries. In this article, we will explore the problem of finding the dimensions that produce the maximum floor area for a rectangular one-story house with a given perimeter.

Problem Statement

The problem can be stated as follows: Find the dimensions of a rectangular one-story house with a perimeter of 131 ft that produce the maximum floor area.

Mathematical Formulation

Let's denote the length of the house as $L$ and the width as $W$. The perimeter of the house is given by the formula:

$$P = 2L + 2W$$

We are given that the perimeter is 131 ft, so we can write:

$$2L + 2W = 131$$

We want to find the values of $L$ and $W$ that maximize the floor area, which is given by the formula:

$$A = LW$$

Solving the Problem

To solve this problem, we can use the method of substitution. We can solve the first equation for $L$ in terms of $W$:

$$L = \frac{131 - 2W}{2}$$

Substituting this expression for $L$ into the second equation, we get:

$$A = \left(\frac{131 - 2W}{2}\right)W$$

Simplifying this expression, we get:

$$A = \frac{131W - 2W^2}{2}$$

To find the maximum value of $A$, we can take the derivative of $A$ with respect to $W$ and set it equal to zero:

$$\frac{dA}{dW} = \frac{131 - 4W}{2} = 0$$

Solving for $W$, we get:

$$W = \frac{131}{4} = 32.75$$

Substituting this value of $W$ back into the expression for $L$, we get:

$$L = \frac{131 - 2(32.75)}{2} = 32.75$$

Therefore, the dimensions that produce the maximum floor area are $L = 32.75$ ft and $W = 32.75$ ft.

Alternative Solution

Another way to solve this problem is to use the method of completing the square. We can rewrite the expression for $A$ as:

$$A = \frac{131W - 2W^2}{2} = \frac{-2(W^2 - 65.5W)}{2} = -W^2 + 65.5W$$

Completing the square, we get:

$$A = -(W - 32.75)^2 + 848.0625$$

This expression is a parabola that opens downward, so the maximum value of $A$ occurs at the vertex, which is at $W = 32.75$ ft.

Conclusion

In this article, we have shown that the dimensions that produce the maximum floor area for a rectangular one-story house with a perimeter of 131 ft are $L = 32.75$ ft and $W = 32.75$ ft. We have used two different methods to solve the problem: the method of substitution and the method of completing the square. Both methods have led to the same solution, which is that the maximum floor area occurs when the length and width of the house are equal.

Comparison with Other Options

To compare our solution with other options, let's consider the following two options:

A. $10.92$ ft $\times 32.75$ ft

B. $3275$ ft $\times 3275$ ft

Option A has a floor area of:

$$A = (10.92)(32.75) = 356.29 \text{ ft}^2$$

Option B has a floor area of:

$$A = (3275)(3275) = 10,703,125 \text{ ft}^2$$

Our solution has a floor area of:

$$A = (32.75)(32.75) = 1073.0625 \text{ ft}^2$$

Therefore, our solution has a floor area that is much larger than option A, but much smaller than option B.

Limitations of the Solution

One limitation of our solution is that it assumes that the perimeter of the house is fixed at 131 ft. In reality, the perimeter of the house may vary depending on the design and construction of the house. Another limitation of our solution is that it assumes that the floor area is the only consideration in designing the house. In reality, other factors such as the cost of construction, the availability of materials, and the aesthetic appeal of the house may also be important considerations.

Future Research Directions

One potential direction for future research is to investigate the effect of varying the perimeter of the house on the floor area. Another potential direction for future research is to investigate the effect of other factors such as the cost of construction and the availability of materials on the design of the house.

Conclusion

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Frequently Asked Questions

Q: What is the maximum floor area for a rectangular one-story house with a perimeter of 131 ft? A: The maximum floor area for a rectangular one-story house with a perimeter of 131 ft is achieved when the length and width of the house are equal, which is 32.75 ft.

Q: How do I calculate the maximum floor area for a rectangular one-story house with a given perimeter? A: To calculate the maximum floor area, you can use the formula A = LW, where L is the length of the house and W is the width of the house. You can also use the method of substitution or the method of completing the square to solve for the maximum floor area.

Q: What are the dimensions of the house that produce the maximum floor area? A: The dimensions of the house that produce the maximum floor area are L = 32.75 ft and W = 32.75 ft.

Q: How does the perimeter of the house affect the maximum floor area? A: The perimeter of the house affects the maximum floor area by limiting the possible values of L and W. A larger perimeter allows for a larger maximum floor area.

Q: What are some other factors that affect the design of a house? A: Some other factors that affect the design of a house include the cost of construction, the availability of materials, and the aesthetic appeal of the house.

Q: How can I optimize the design of a house to achieve the maximum floor area? A: To optimize the design of a house to achieve the maximum floor area, you can use mathematical techniques such as the method of substitution or the method of completing the square. You can also consider factors such as the cost of construction and the availability of materials.

Q: What are some potential applications of optimizing floor area for a rectangular one-story house? A: Some potential applications of optimizing floor area for a rectangular one-story house include designing houses for maximum efficiency, minimizing construction costs, and creating aesthetically pleasing designs.

Q: Can I use the same techniques to optimize the design of other types of buildings? A: Yes, the techniques used to optimize the design of a rectangular one-story house can be applied to other types of buildings, such as apartments, offices, and warehouses.

Q: What are some potential limitations of optimizing floor area for a rectangular one-story house? A: Some potential limitations of optimizing floor area for a rectangular one-story house include the assumption that the perimeter of the house is fixed, the assumption that the floor area is the only consideration in designing the house, and the potential for conflicting design goals.

Q: How can I further research the topic of optimizing floor area for a rectangular one-story house? A: You can further research the topic of optimizing floor area for a rectangular one-story house by reading books and articles on the subject, attending conferences and workshops, and consulting with experts in the field.

Conclusion

In conclusion, we have provided a Q&A article on optimizing floor area for a rectangular one-story house. We have answered common questions on the topic, including how to calculate the maximum floor area, the dimensions of the house that produce the maximum floor area, and how the perimeter of the house affects the maximum floor area. We have also discussed potential applications and limitations of optimizing floor area for a rectangular one-story house.