Select The Correct Answer.Raymond Wants To Increase The Size Of His Fenced Yard For The Dog He Plans To Adopt Next Month. He Wants The Area Of The Fenced Yard To Be 900 Square Meters. If The Resulting Width Of The Fenced Yard Is 11 Meters Less Than The
Introduction
As a responsible dog owner, Raymond wants to ensure that his new furry friend has a spacious and secure area to play and exercise. With a target area of 900 square meters, he needs to determine the optimal dimensions of his fenced yard. In this article, we will explore the mathematical concepts involved in increasing the area of a fenced yard and provide a step-by-step solution to help Raymond achieve his goal.
Understanding the Problem
Raymond wants to increase the size of his fenced yard to accommodate his new dog. The area of the fenced yard should be 900 square meters. Let's denote the width of the fenced yard as w and the length as l. The area of a rectangle (such as a fenced yard) is given by the formula:
Area = length × width
In this case, the area is 900 square meters, so we can write:
900 = l × w
The Relationship Between Width and Length
The problem states that the resulting width of the fenced yard is 11 meters less than the length. We can express this relationship as:
w = l - 11
Substituting the Relationship into the Area Formula
Now that we have a relationship between the width and length, we can substitute the expression for w into the area formula:
900 = l × (l - 11)
Expanding and Simplifying the Equation
To solve for l, we need to expand and simplify the equation:
900 = l^2 - 11l
Rearranging the terms, we get:
l^2 - 11l - 900 = 0
Solving the Quadratic Equation
This is a quadratic equation in the form ax^2 + bx + c = 0, where a = 1, b = -11, and c = -900. We can solve for l using the quadratic formula:
l = (-b ± √(b^2 - 4ac)) / 2a
Plugging in the values, we get:
l = (11 ± √((-11)^2 - 4(1)(-900))) / 2(1)
l = (11 ± √(121 + 3600)) / 2
l = (11 ± √3721) / 2
l = (11 ± 61) / 2
Finding the Two Possible Values of Length
Now we have two possible values for l:
l = (11 + 61) / 2 = 72 / 2 = 36
l = (11 - 61) / 2 = -50 / 2 = -25
Since the length cannot be negative, we discard the second solution.
Finding the Corresponding Width
Now that we have the length, we can find the corresponding width using the relationship:
w = l - 11
w = 36 - 11
w = 25
Conclusion
In this article, we have used mathematical concepts to help Raymond determine the optimal dimensions of his fenced yard. By understanding the relationship between the width and length, we were able to solve for the length and find the corresponding width. The resulting dimensions are a length of 36 meters and a width of 25 meters, which will provide a secure and spacious area for Raymond's new dog to play and exercise.
Additional Considerations
While this article has provided a mathematical solution to the problem, there are additional considerations that Raymond should take into account when designing his fenced yard. These include:
- Safety features: Raymond should ensure that the fenced yard is safe for his dog to play and exercise. This may include features such as a secure gate, a non-slip surface, and adequate drainage.
- Aesthetics: Raymond should consider the visual appeal of the fenced yard and ensure that it complements the surrounding landscape.
- Maintenance: Raymond should consider the maintenance requirements of the fenced yard, including regular cleaning and repairs.
Q: What is the optimal shape for a fenced yard?
A: The optimal shape for a fenced yard depends on the specific needs and goals of the owner. However, a rectangular shape is often the most practical and efficient choice, as it allows for a larger area while minimizing the perimeter.
Q: How do I calculate the area of a fenced yard?
A: To calculate the area of a fenced yard, you need to multiply the length and width of the yard. The formula for the area of a rectangle is:
Area = length × width
Q: What is the relationship between the width and length of a fenced yard?
A: In the case of Raymond's fenced yard, the width is 11 meters less than the length. This relationship can be expressed as:
w = l - 11
Q: How do I solve for the length of a fenced yard?
A: To solve for the length of a fenced yard, you need to use the quadratic formula:
l = (-b ± √(b^2 - 4ac)) / 2a
where a = 1, b = -11, and c = -900. Plugging in these values, you get:
l = (11 ± √((-11)^2 - 4(1)(-900))) / 2(1)
l = (11 ± √(121 + 3600)) / 2
l = (11 ± √3721) / 2
l = (11 ± 61) / 2
Q: What are the two possible values of length for a fenced yard?
A: The two possible values of length for a fenced yard are:
l = (11 + 61) / 2 = 72 / 2 = 36
l = (11 - 61) / 2 = -50 / 2 = -25
Since the length cannot be negative, we discard the second solution.
Q: How do I find the corresponding width of a fenced yard?
A: To find the corresponding width of a fenced yard, you need to use the relationship:
w = l - 11
Plugging in the value of l, you get:
w = 36 - 11
w = 25
Q: What are some additional considerations when designing a fenced yard?
A: Some additional considerations when designing a fenced yard include:
- Safety features: Ensure that the fenced yard is safe for pets and children to play and exercise.
- Aesthetics: Consider the visual appeal of the fenced yard and ensure that it complements the surrounding landscape.
- Maintenance: Consider the maintenance requirements of the fenced yard, including regular cleaning and repairs.
Q: How can I ensure that my fenced yard is safe for my pets?
A: To ensure that your fenced yard is safe for your pets, you should:
- Use secure fencing: Use fencing that is sturdy and secure to prevent pets from escaping.
- Install safety features: Install safety features such as gates, latches, and locks to prevent pets from escaping.
- Supervise pets: Supervise pets when they are in the fenced yard to ensure that they are safe and secure.
Q: How can I make my fenced yard more aesthetically pleasing?
A: To make your fenced yard more aesthetically pleasing, you can:
- Use decorative fencing: Use decorative fencing such as picket fencing or ornamental fencing to add visual appeal.
- Add landscaping: Add landscaping such as plants, flowers, and trees to add visual appeal.
- Use outdoor lighting: Use outdoor lighting to highlight the features of the fenced yard and add visual appeal.
Q: How can I maintain my fenced yard?
A: To maintain your fenced yard, you should:
- Regularly clean the yard: Regularly clean the yard to remove debris and dirt.
- Repair damaged fencing: Repair damaged fencing to prevent further damage.
- Trim plants and trees: Trim plants and trees to maintain their shape and size.