Finding the Dimensions of the New Lot

Oct 13, 2025 by windailys.com 91 views

Introduction

In mathematics, the perimeter and area of a rectangle are two fundamental concepts that are often used to solve problems. The perimeter of a rectangle is the total length of its boundary, while the area is the amount of space inside the rectangle. In this article, we will explore how to find the dimensions of a rectangular lot that has the same perimeter as another lot but a smaller area.

Understanding the Problem

Let's consider a rectangular lot that is 55 yards wide and 90 yards long. We are asked to find the dimensions of another rectangular lot that has the same perimeter but a smaller area. To solve this problem, we need to understand the relationship between the perimeter and area of a rectangle.

Perimeter of a Rectangle

The perimeter of a rectangle is given by the formula:

P = 2(l + w)

where P is the perimeter, l is the length, and w is the width.

Area of a Rectangle

The area of a rectangle is given by the formula:

A = l × w

where A is the area, l is the length, and w is the width.

Finding the Dimensions of the New Lot

To find the dimensions of the new lot, we need to find the length and width that will give us the same perimeter but a smaller area. Let's assume that the new lot has a width of x yards and a length of y yards.

The perimeter of the new lot is given by:

P = 2(x + y)

Since the perimeter of the new lot is the same as the original lot, we can set up the equation:

2(x + y) = 2(55 + 90)

Simplifying the equation, we get:

x + y = 145

Now, we need to find the values of x and y that will give us a smaller area. We can do this by substituting the values of x and y into the area formula:

A = xy

We want to find the values of x and y that will give us the smallest possible area.

Using Algebra to Find the Dimensions

To find the dimensions of the new lot, we can use algebra to solve the equation:

x + y = 145

We can rewrite the equation as:

y = 145 - x

Substituting this expression for y into the area formula, we get:

A = x(145 - x)

Expanding the equation, we get:

A = 145x - x^2

To find the values of x that will give us the smallest possible area, we can take the derivative of the area equation with respect to x:

dA/dx = 145 - 2x

Setting the derivative equal to zero, we get:

145 - 2x = 0

Solving for x, we get:

x = 72.5

Now that we have found the value of x, we can find the value of y by substituting x into the equation:

y = 145 - x

y = 145 - 72.5

y = 72.5

Conclusion

In this article, we have explored how to find the dimensions of a rectangular lot that has the same perimeter but a smaller area. We used algebra to solve the equation and found that the new lot has a width of 72.5 yards and a length of 72.5 yards.

The Final Answer

Introduction

In our previous article, we explored how to find the dimensions of a rectangular lot that has the same perimeter but a smaller area. We used algebra to solve the equation and found that the new lot has a width of 72.5 yards and a length of 72.5 yards. In this article, we will answer some of the most frequently asked questions about this problem.

Q: What is the perimeter of the original lot?

A: The perimeter of the original lot is given by the formula:

P = 2(l + w)

where P is the perimeter, l is the length, and w is the width. In this case, the length is 90 yards and the width is 55 yards, so the perimeter is:

P = 2(90 + 55) = 2(145) = 290 yards

Q: What is the area of the original lot?

A: The area of the original lot is given by the formula:

A = l × w

where A is the area, l is the length, and w is the width. In this case, the length is 90 yards and the width is 55 yards, so the area is:

A = 90 × 55 = 4950 square yards

Q: How did you find the dimensions of the new lot?

A: We used algebra to solve the equation:

x + y = 145

where x is the width and y is the length of the new lot. We also used the area formula:

A = xy

to find the values of x and y that will give us the smallest possible area.

Q: Why did you choose the value of x = 72.5?

A: We chose the value of x = 72.5 because it is the value that will give us the smallest possible area. We found this value by taking the derivative of the area equation with respect to x and setting it equal to zero.

Q: What is the area of the new lot?

A: The area of the new lot is given by the formula:

A = xy

where A is the area, x is the width, and y is the length. In this case, the width is 72.5 yards and the length is 72.5 yards, so the area is:

A = 72.5 × 72.5 = 5256.25 square yards

Q: How does the new lot compare to the original lot?

A: The new lot has a smaller area than the original lot. The original lot has an area of 4950 square yards, while the new lot has an area of 5256.25 square yards. This means that the new lot is slightly larger than the original lot.

Q: What are some real-world applications of this problem?

A: This problem has many real-world applications, such as:

Finding the dimensions of a rectangular garden or park that has the same perimeter but a smaller area.
Designing a rectangular building or room that has the same perimeter but a smaller area.
Optimizing the layout of a rectangular factory or warehouse to maximize space.

Conclusion

In this article, we have answered some of the most frequently asked questions about finding the dimensions of a rectangular lot that has the same perimeter but a smaller area. We hope that this article has been helpful in understanding this problem and its applications.