Problem 1: Fencing Around A PoolYou Are Putting Up A Fence Around Your New Pool. The Length Is Four Feet Less Than Twice The Width. 1. Sketch And Label A Picture To Represent The Scenario. (Dimensions Are In Feet.)2. Write A Simplified Polynomial
Understanding the Scenario
When it comes to fencing around a pool, it's essential to consider the dimensions of the area to be enclosed. In this problem, we are given that the length of the fence is four feet less than twice the width. This creates a relationship between the length and width of the pool, which we can represent mathematically.
Sketching and Labeling a Picture
To visualize the scenario, let's sketch a picture of the pool with the fence. We can represent the width of the pool as w and the length as l. Since the length is four feet less than twice the width, we can write an equation to represent this relationship:
l = 2w - 4
We can also label the picture to show the dimensions of the pool. For example:
- Width (w): _______
- Length (l): _______
Writing a Simplified Polynomial
To write a simplified polynomial, we need to express the relationship between the length and width of the pool in terms of a single variable. Let's choose the width (w) as the variable. We can substitute the expression for the length (l = 2w - 4) into the equation:
l = 2w - 4
Since we want to express the length in terms of the width, we can rewrite the equation as:
2w - 4 = l
Now, we can substitute this expression for the length into the equation for the area of the pool:
Area = length × width = (2w - 4) × w
Expanding the equation, we get:
Area = 2w^2 - 4w
This is a simplified polynomial that represents the relationship between the area of the pool and the width.
Simplifying the Polynomial
To simplify the polynomial, we can combine like terms. In this case, there are no like terms to combine, so the polynomial is already simplified.
Interpreting the Polynomial
The polynomial 2w^2 - 4w represents the area of the pool in terms of the width. This means that if we know the width of the pool, we can use this polynomial to calculate the area.
Example
Let's say the width of the pool is 10 feet. We can substitute this value into the polynomial to calculate the area:
Area = 2(10)^2 - 4(10) = 2(100) - 40 = 200 - 40 = 160
So, the area of the pool is 160 square feet.
Conclusion
In this problem, we used algebraic techniques to represent the relationship between the length and width of a pool. We wrote a simplified polynomial to express the area of the pool in terms of the width. This polynomial can be used to calculate the area of the pool given the width.
Key Takeaways
- The length of the fence is four feet less than twice the width.
- The area of the pool can be represented by the polynomial 2w^2 - 4w.
- The polynomial can be used to calculate the area of the pool given the width.
Further Exploration
- What if the width of the pool is not a fixed value? How would this affect the polynomial?
- Can we use the polynomial to calculate the length of the fence given the width?
- What if the pool is not a rectangle? How would this affect the polynomial?
References
Related Problems
Glossary
- Width: The distance between the two sides of the pool.
- Length: The distance between the two ends of the pool.
- Area: The amount of space inside the pool.
- Polynomial: An algebraic expression consisting of variables and coefficients.
Frequently Asked Questions
Q: What is the relationship between the length and width of the pool?
A: The length of the pool is four feet less than twice the width. This can be represented by the equation: l = 2w - 4
Q: How do I calculate the area of the pool?
A: To calculate the area of the pool, you can use the polynomial 2w^2 - 4w, where w is the width of the pool.
Q: What if the width of the pool is not a fixed value?
A: If the width of the pool is not a fixed value, you can use the polynomial 2w^2 - 4w to calculate the area of the pool for any given width.
Q: Can I use the polynomial to calculate the length of the fence given the width?
A: Yes, you can use the equation l = 2w - 4 to calculate the length of the fence given the width.
Q: What if the pool is not a rectangle?
A: If the pool is not a rectangle, the polynomial 2w^2 - 4w will not be accurate. You will need to use a different method to calculate the area of the pool.
Q: How do I simplify the polynomial?
A: To simplify the polynomial, you can combine like terms. In this case, there are no like terms to combine, so the polynomial is already simplified.
Q: What is the significance of the polynomial 2w^2 - 4w?
A: The polynomial 2w^2 - 4w represents the area of the pool in terms of the width. This means that if you know the width of the pool, you can use this polynomial to calculate the area.
Q: Can I use the polynomial to solve real-world problems?
A: Yes, you can use the polynomial to solve real-world problems involving the area of a pool. For example, you can use the polynomial to calculate the amount of fencing needed to enclose a pool.
Additional Questions and Answers
Q: What is the formula for the area of a rectangle?
A: The formula for the area of a rectangle is A = length × width.
Q: How do I calculate the perimeter of a rectangle?
A: To calculate the perimeter of a rectangle, you can use the formula P = 2(length + width).
Q: What is the difference between the area and the perimeter of a rectangle?
A: The area of a rectangle is the amount of space inside the rectangle, while the perimeter is the distance around the rectangle.
Conclusion
In this Q&A article, we have answered some of the most frequently asked questions about the problem of fencing around a pool. We have covered topics such as the relationship between the length and width of the pool, how to calculate the area of the pool, and how to simplify the polynomial. We have also provided additional questions and answers to help you better understand the problem.
Key Takeaways
- The length of the pool is four feet less than twice the width.
- The area of the pool can be represented by the polynomial 2w^2 - 4w.
- The polynomial can be used to calculate the area of the pool given the width.
- The formula for the area of a rectangle is A = length × width.
- The formula for the perimeter of a rectangle is P = 2(length + width).
Further Exploration
- What if the pool is not a rectangle? How would this affect the polynomial?
- Can you use the polynomial to solve real-world problems involving the area of a pool?
- What is the significance of the polynomial 2w^2 - 4w?
References
Related Problems
Glossary
- Width: The distance between the two sides of the pool.
- Length: The distance between the two ends of the pool.
- Area: The amount of space inside the pool.
- Polynomial: An algebraic expression consisting of variables and coefficients.
- Perimeter: The distance around the pool.