The Formulas for Perimeter and Area

Oct 13, 2025 by thesweetdesigns.com 138 views

Introduction

When it comes to designing a parking lot, understanding the dimensions is crucial for ensuring that it can accommodate the required number of vehicles. In this article, we will delve into the world of mathematics and explore how to find the dimensions of a rectangular parking lot given its perimeter and area. We will use the formulas for perimeter and area to solve for the length and width of the parking lot.

The Formulas for Perimeter and Area

The perimeter of a rectangle is given by the formula:

P = 2l + 2w

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

The area of a rectangle is given by the formula:

A = lw

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

Given Information

We are given that the perimeter of the parking lot is 820 ft and the area is 42,000 ft^2. We can use this information to set up a system of equations.

Setting Up the System of Equations

Using the formula for perimeter, we can write:

2l + 2w = 820

Using the formula for area, we can write:

lw = 42,000

Solving the System of Equations

We can start by simplifying the first equation by dividing both sides by 2:

l + w = 410

Next, we can solve for one variable in terms of the other. Let's solve for w in terms of l:

w = 410 - l

Now, we can substitute this expression for w into the second equation:

l(410 - l) = 42,000

Expanding the left-hand side, we get:

410l - l^2 = 42,000

Rearranging the equation to form a quadratic equation, we get:

l^2 - 410l + 42,000 = 0

Solving the Quadratic Equation

We can solve this quadratic equation using the quadratic formula:

l = (-b ± √(b^2 - 4ac)) / 2a

In this case, a = 1, b = -410, and c = 42,000. Plugging these values into the formula, we get:

l = (410 ± √((-410)^2 - 4(1)(42,000))) / 2(1)

Simplifying the expression under the square root, we get:

l = (410 ± √(169,100 - 168,000)) / 2

l = (410 ± √1,100) / 2

l = (410 ± 33.33) / 2

Finding the Length and Width

We can now find the two possible values for l:

l = (410 + 33.33) / 2 = 371.67

l = (410 - 33.33) / 2 = 188.33

Now that we have found the two possible values for l, we can find the corresponding values for w:

w = 410 - l

w = 410 - 371.67 = 38.33

w = 410 - 188.33 = 221.67

Checking the Solutions

We can check our solutions by plugging them back into the original equations. Let's check the first solution:

l = 371.67

w = 38.33

2l + 2w = 2(371.67) + 2(38.33) = 741.33 + 76.66 = 818

lw = (371.67)(38.33) = 14,200.11

Since this solution does not satisfy the original equations, we can discard it.

Conclusion

The only solution that satisfies the original equations is:

l = 188.33

w = 221.67

Therefore, the dimensions of the parking lot are approximately 188.33 ft by 221.67 ft.

Discussion

The problem of finding the dimensions of a rectangular parking lot given its perimeter and area is a classic example of a system of equations. By using the formulas for perimeter and area, we can set up a system of equations and solve for the length and width of the parking lot. In this article, we used the quadratic formula to solve for the length and width, and we found that the dimensions of the parking lot are approximately 188.33 ft by 221.67 ft.

Final Answer

The final answer is: $\boxed{188.33}$

Introduction

In our previous article, we explored how to find the dimensions of a rectangular parking lot given its perimeter and area. We used the formulas for perimeter and area to set up a system of equations and solve for the length and width of the parking lot. In this article, we will answer some frequently asked questions related to this topic.

Q&A

Q: What is the formula for the perimeter of a rectangle?

A: The formula for the perimeter of a rectangle is P = 2l + 2w, where P is the perimeter, l is the length, and w is the width.

Q: What is the formula for the area of a rectangle?

A: The formula for the area of a rectangle is A = lw, where A is the area, l is the length, and w is the width.

Q: How do I set up a system of equations to find the dimensions of a rectangular parking lot given its perimeter and area?

A: To set up a system of equations, you can use the formulas for perimeter and area. For example, if the perimeter is 820 ft and the area is 42,000 ft^2, you can write:

2l + 2w = 820

lw = 42,000

Q: How do I solve a system of equations?

A: There are several methods to solve a system of equations, including substitution, elimination, and graphing. In the case of the parking lot problem, we used the quadratic formula to solve for the length and width.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that is used to solve quadratic equations. It is given by:

x = (-b ± √(b^2 - 4ac)) / 2a

Q: How do I use the quadratic formula to solve for the length and width of a rectangular parking lot?

A: To use the quadratic formula, you need to plug in the values of a, b, and c into the formula. In the case of the parking lot problem, a = 1, b = -410, and c = 42,000. Plugging these values into the formula, we get:

l = (410 ± √((-410)^2 - 4(1)(42,000))) / 2(1)

Simplifying the expression under the square root, we get:

l = (410 ± √(169,100 - 168,000)) / 2

l = (410 ± √1,100) / 2

l = (410 ± 33.33) / 2

Q: What are the dimensions of the parking lot?

A: The dimensions of the parking lot are approximately 188.33 ft by 221.67 ft.

Q: How do I check my solutions?

A: To check your solutions, you can plug them back into the original equations. If the solutions satisfy the original equations, then they are correct.

Conclusion

In this article, we answered some frequently asked questions related to finding the dimensions of a rectangular parking lot given its perimeter and area. We used the formulas for perimeter and area to set up a system of equations and solve for the length and width of the parking lot. We also used the quadratic formula to solve for the length and width. We hope that this article has been helpful in answering your questions.

Final Answer

The final answer is: $\boxed{188.33}$