A Rectangular Swimming Pool Has A Perimeter Of 96 Ft. The Area Of The Pool Is $504 \, \text{ft}^2$. Which System Of Equations Models This Situation Correctly, Where $l$ Is The Length Of The Pool In Feet And $w$ Is The Width

Introduction

When it comes to designing and building a swimming pool, understanding the relationship between its perimeter and area is crucial. In this article, we will explore how to model a rectangular swimming pool using a system of equations, where the perimeter and area are given. We will use the given information to determine the correct system of equations that represents this situation.

The Problem

A rectangular swimming pool has a perimeter of 96 ft. The area of the pool is $504 \, \text{ft}^2$. We are asked to find the system of equations that models this situation correctly, where $l$ is the length of the pool in feet and $w$ is the width.

Understanding the Perimeter and Area of a Rectangle

The perimeter of a rectangle is given by the formula $P = 2l + 2w$, where $l$ is the length and $w$ is the width. The area of a rectangle is given by the formula $A = lw$.

Given Information

We are given that the perimeter of the pool is 96 ft, so we can write an equation based on this information:

$$2l + 2w = 96$$

We are also given that the area of the pool is $504 \, \text{ft}^2$, so we can write another equation based on this information:

$$lw = 504$$

System of Equations

We now have two equations that represent the perimeter and area of the pool:

$$2l + 2w = 96$$

$$lw = 504$$

These two equations form a system of equations that models the situation correctly.

Solving the System of Equations

To solve this system of equations, we can use substitution or elimination. Let's use substitution. We can solve the first equation for $l$ in terms of $w$:

$$2l = 96 - 2w$$

$$l = 48 - w$$

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

$$(48 - w)w = 504$$

Expanding and simplifying, we get:

$$48w - w^2 = 504$$

Rearranging, we get:

$$w^2 - 48w + 504 = 0$$

This is a quadratic equation in $w$. We can solve it using the quadratic formula:

$$w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

In this case, $a = 1$, $b = -48$, and $c = 504$. Plugging these values into the formula, we get:

$$w = \frac{48 \pm \sqrt{(-48)^2 - 4(1)(504)}}{2(1)}$$

Simplifying, we get:

$$w = \frac{48 \pm \sqrt{2304 - 2016}}{2}$$

$$w = \frac{48 \pm \sqrt{288}}{2}$$

$$w = \frac{48 \pm 12\sqrt{2}}{2}$$

$$w = 24 \pm 6\sqrt{2}$$

Finding the Length

Now that we have found the width, we can find the length by substituting the expression for $w$ into one of the original equations. Let's use the first equation:

$$2l + 2(24 \pm 6\sqrt{2}) = 96$$

Simplifying, we get:

$$2l + 48 \pm 12\sqrt{2} = 96$$

Subtracting 48 from both sides, we get:

$$2l \pm 12\sqrt{2} = 48$$

Dividing both sides by 2, we get:

$$l \pm 6\sqrt{2} = 24$$

Conclusion

In this article, we have modeled a rectangular swimming pool using a system of equations, where the perimeter and area are given. We have used the given information to determine the correct system of equations that represents this situation. We have also solved the system of equations to find the length and width of the pool.

Final Answer

The final answer is:

$$l = 24 \pm 6\sqrt{2}$$

$$w = 24 \pm 6\sqrt{2}$$

Note: The final answer is the same for both the length and width, as the solution is a pair of values that satisfy both equations.

Introduction

In our previous article, we explored how to model a rectangular swimming pool using a system of equations, where the perimeter and area are given. We used the given information to determine the correct system of equations that represents this situation. In this article, we will answer some frequently asked questions related to this topic.

Q: What is the perimeter of a rectangle?

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

Q: What is the area of a rectangle?

A: The area of a rectangle is given by the formula $A = lw$.

Q: How do I find the length and width of a rectangle if I know its perimeter and area?

A: To find the length and width of a rectangle, you can use the following system of equations:

$$2l + 2w = P$$

$$lw = A$$

where $P$ is the perimeter and $A$ is the area.

Q: Can I use substitution or elimination to solve the system of equations?

A: Yes, you can use either substitution or elimination to solve the system of equations. Let's use substitution. We can solve the first equation for $l$ in terms of $w$:

$$2l = P - 2w$$

$$l = \frac{P - 2w}{2}$$

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

$$\left(\frac{P - 2w}{2}\right)w = A$$

Expanding and simplifying, we get:

$$Pw - 2w^2 = 2A$$

Rearranging, we get:

$$2w^2 - Pw + 2A = 0$$

This is a quadratic equation in $w$. We can solve it using the quadratic formula:

$$w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

In this case, $a = 2$, $b = -P$, and $c = 2A$. Plugging these values into the formula, we get:

$$w = \frac{P \pm \sqrt{(-P)^2 - 4(2)(2A)}}{2(2)}$$

Simplifying, we get:

$$w = \frac{P \pm \sqrt{P^2 - 16A}}{4}$$

Q: What if I have a quadratic equation in the form $ax^2 + bx + c = 0$?

A: If you have a quadratic equation in the form $ax^2 + bx + c = 0$, you can use the quadratic formula to solve it:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

Q: Can I use a calculator to solve the quadratic equation?

A: Yes, you can use a calculator to solve the quadratic equation. Simply plug in the values of $a$, $b$, and $c$ into the quadratic formula and solve for $x$.

Q: What if I have a system of equations with more than two variables?

A: If you have a system of equations with more than two variables, you can use substitution or elimination to solve it. However, you may need to use a more advanced method, such as Gaussian elimination or matrix operations.

Conclusion

In this article, we have answered some frequently asked questions related to modeling a rectangular swimming pool using a system of equations. We have also provided some tips and tricks for solving quadratic equations and systems of equations.

Final Answer

The final answer is:

$$l = \frac{P - 2w}{2}$$

$$w = \frac{P \pm \sqrt{P^2 - 16A}}{4}$$

Note: The final answer is the same for both the length and width, as the solution is a pair of values that satisfy both equations.