A School Is Constructing A Rectangular Play Area Against An Exterior Wall Of The School Building. It Uses 120 Feet Of Fencing Material To Enclose Three Sides Of The Play Area.Complete The Table By Giving The Length And Area For Each Width. (The Width
A Real-World Application of Mathematics: Finding the Dimensions of a Rectangular Play Area
In this article, we will explore a real-world problem that involves the use of mathematical concepts to find the dimensions of a rectangular play area. The problem is as follows: a school is constructing a rectangular play area against an exterior wall of the school building. It uses 120 feet of fencing material to enclose three sides of the play area. We will use mathematical formulas to find the length and area of the play area for each possible width.
- A school is constructing a rectangular play area against an exterior wall of the school building.
- It uses 120 feet of fencing material to enclose three sides of the play area.
- We need to find the length and area of the play area for each possible width.
To solve this problem, we will use the following mathematical formulas:
- Perimeter of a rectangle: P = 2l + 2w, where P is the perimeter, l is the length, and w is the width.
- Area of a rectangle: A = lw, where A is the area, l is the length, and w is the width.
Let's assume that the width of the play area is w feet. Since the school uses 120 feet of fencing material to enclose three sides of the play area, the perimeter of the play area is 120 feet. We can set up an equation using the perimeter formula:
2l + 2w = 120
Since the play area is against an exterior wall of the school building, one side of the play area is not fenced. Therefore, we can assume that the length of the play area is equal to the width of the play area plus the width of the unfenced side. Let's call the width of the unfenced side x. Then, the length of the play area is l = w + x.
Substituting this expression for l into the perimeter equation, we get:
2(w + x) + 2w = 120
Simplifying the equation, we get:
2w + 2x + 2w = 120
Combine like terms:
4w + 2x = 120
Subtract 2x from both sides:
4w = 120 - 2x
Divide both sides by 4:
w = (120 - 2x) / 4
Simplify the expression:
w = 30 - 0.5x
Now, we can find the length of the play area by substituting the expression for w into the equation l = w + x:
l = (30 - 0.5x) + x
Simplify the expression:
l = 30 + 0.5x
Now, we can find the area of the play area by substituting the expressions for l and w into the area formula:
A = lw = (30 + 0.5x)(30 - 0.5x)
Expand the expression:
A = 900 - 0.25x^2
Now, we can find the length and area of the play area for each possible width by substituting different values of x into the expressions for l and A.
| Width (w) | Length (l) | Area (A) |
|---|---|---|
| 10 | 35 | 350 |
| 20 | 40 | 800 |
| 30 | 45 | 1350 |
| 40 | 50 | 2000 |
| 50 | 55 | 2750 |
| 60 | 60 | 3600 |
| 70 | 65 | 4550 |
| 80 | 70 | 5600 |
| 90 | 75 | 6750 |
| 100 | 80 | 8000 |
In this article, we used mathematical formulas to find the length and area of a rectangular play area for each possible width. We assumed that the school uses 120 feet of fencing material to enclose three sides of the play area and that the length of the play area is equal to the width of the play area plus the width of the unfenced side. We found the length and area of the play area by substituting different values of x into the expressions for l and A. The results are shown in the table above.
This problem has many real-world applications. For example, architects and engineers use mathematical formulas to design buildings and other structures. They need to consider the perimeter and area of the structure to ensure that it is safe and functional. This problem also has applications in agriculture, where farmers need to calculate the area of their fields to determine how much fertilizer and water to apply.
There are many future research directions for this problem. For example, we could investigate the effect of different shapes on the perimeter and area of the play area. We could also explore the use of different materials for the fencing, such as wood or metal. Additionally, we could investigate the impact of the play area on the surrounding environment, such as the effect on local wildlife.
There are several limitations to this study. For example, we assumed that the school uses 120 feet of fencing material to enclose three sides of the play area. In reality, the school may use more or less fencing material, depending on the design of the play area. We also assumed that the length of the play area is equal to the width of the play area plus the width of the unfenced side. In reality, the length of the play area may be different, depending on the design of the play area.
Based on the results of this study, we recommend the following for future research:
- Investigate the effect of different shapes on the perimeter and area of the play area.
- Explore the use of different materials for the fencing, such as wood or metal.
- Investigate the impact of the play area on the surrounding environment, such as the effect on local wildlife.
- Consider the effect of different designs on the perimeter and area of the play area.
In conclusion, this study used mathematical formulas to find the length and area of a rectangular play area for each possible width. We assumed that the school uses 120 feet of fencing material to enclose three sides of the play area and that the length of the play area is equal to the width of the play area plus the width of the unfenced side. The results are shown in the table above. We recommend future research in the areas of different shapes, materials, and designs.
A School is Constructing a Rectangular Play Area: Q&A
In our previous article, we explored a real-world problem that involved the use of mathematical concepts to find the dimensions of a rectangular play area. The problem was as follows: a school is constructing a rectangular play area against an exterior wall of the school building. It uses 120 feet of fencing material to enclose three sides of the play area. We used mathematical formulas to find the length and area of the play area for each possible width.
Q: What is the perimeter of the play area? A: The perimeter of the play area is 120 feet.
Q: How do we find the length of the play area? A: We find the length of the play area by substituting the expression for w into the equation l = w + x.
Q: What is the expression for w? A: The expression for w is w = 30 - 0.5x.
Q: How do we find the area of the play area? A: We find the area of the play area by substituting the expressions for l and w into the area formula: A = lw.
Q: What is the formula for the area of a rectangle? A: The formula for the area of a rectangle is A = lw.
Q: How do we find the length and area of the play area for each possible width? A: We find the length and area of the play area for each possible width by substituting different values of x into the expressions for l and A.
Q: What are the results of the calculations? A: The results of the calculations are shown in the table below:
| Width (w) | Length (l) | Area (A) |
|---|---|---|
| 10 | 35 | 350 |
| 20 | 40 | 800 |
| 30 | 45 | 1350 |
| 40 | 50 | 2000 |
| 50 | 55 | 2750 |
| 60 | 60 | 3600 |
| 70 | 65 | 4550 |
| 80 | 70 | 5600 |
| 90 | 75 | 6750 |
| 100 | 80 | 8000 |
Q: What are the implications of this study? A: The implications of this study are that architects and engineers can use mathematical formulas to design buildings and other structures. They need to consider the perimeter and area of the structure to ensure that it is safe and functional. This problem also has applications in agriculture, where farmers need to calculate the area of their fields to determine how much fertilizer and water to apply.
Q: What are the limitations of this study? A: The limitations of this study are that we assumed that the school uses 120 feet of fencing material to enclose three sides of the play area. In reality, the school may use more or less fencing material, depending on the design of the play area. We also assumed that the length of the play area is equal to the width of the play area plus the width of the unfenced side. In reality, the length of the play area may be different, depending on the design of the play area.
Q: What are the recommendations for future research? A: The recommendations for future research are:
- Investigate the effect of different shapes on the perimeter and area of the play area.
- Explore the use of different materials for the fencing, such as wood or metal.
- Investigate the impact of the play area on the surrounding environment, such as the effect on local wildlife.
- Consider the effect of different designs on the perimeter and area of the play area.
In conclusion, this Q&A article provides answers to common questions about the problem of finding the dimensions of a rectangular play area. We used mathematical formulas to find the length and area of the play area for each possible width. The results are shown in the table above. We recommend future research in the areas of different shapes, materials, and designs.