The Length Of A Basketball Field Is 12 Meters Greater Than Its Wide, If The Perimeter Is 96 Meters, What Are The Dimensions Of The Field?
Introduction
In this problem, we are given the relationship between the length and width of a basketball field, as well as its perimeter. We need to use this information to find the dimensions of the field. The perimeter of a rectangle, such as a basketball field, is the sum of the lengths of all its sides. In this case, the perimeter is given as 96 meters.
Understanding the Relationship Between Length and Width
Let's denote the width of the field as w and the length as l. We are told that the length is 12 meters greater than the width, so we can write an equation:
l = w + 12
Understanding the Perimeter of the Field
The perimeter of a rectangle is given by the formula:
P = 2l + 2w
We are given that the perimeter is 96 meters, so we can substitute this value into the formula:
96 = 2l + 2w
Substituting the Relationship Between Length and Width
Now, we can substitute the equation l = w + 12 into the perimeter equation:
96 = 2(w + 12) + 2w
Simplifying the Equation
Expanding the equation, we get:
96 = 2w + 24 + 2w
Combine like terms:
96 = 4w + 24
Solving for Width
Subtract 24 from both sides:
72 = 4w
Divide both sides by 4:
w = 18
Finding the Length
Now that we have the width, we can find the length by substituting the value of w into the equation l = w + 12:
l = 18 + 12
l = 30
Conclusion
In this problem, we used the relationship between the length and width of a basketball field, as well as its perimeter, to find the dimensions of the field. We found that the width is 18 meters and the length is 30 meters.
Example Use Case
This problem can be used to demonstrate the concept of algebraic equations and how they can be used to solve real-world problems. It can also be used to practice solving systems of linear equations.
Tips and Tricks
- When solving equations, make sure to follow the order of operations (PEMDAS).
- Use substitution and elimination methods to solve systems of linear equations.
- Practice, practice, practice! The more you practice solving equations, the more comfortable you will become with the process.
Common Mistakes
- Failing to follow the order of operations (PEMDAS).
- Not using substitution and elimination methods to solve systems of linear equations.
- Not checking the solution to make sure it is reasonable.
Real-World Applications
This problem has real-world applications in fields such as architecture, engineering, and construction. It can be used to design and build structures such as buildings, bridges, and roads.
Future Directions
This problem can be extended to more complex problems, such as finding the dimensions of a field with a non-rectangular shape. It can also be used to explore more advanced topics in mathematics, such as calculus and differential equations.
Conclusion
In conclusion, this problem demonstrates the importance of algebraic equations in solving real-world problems. It requires the use of substitution and elimination methods to solve systems of linear equations, and it has real-world applications in fields such as architecture, engineering, and construction.
Q&A: The length of a basketball field is 12 meters greater than its wide, if the perimeter is 96 meters, what are the dimensions of the field?
Q: What is the relationship between the length and width of the basketball field?
A: The length of the basketball field is 12 meters greater than its width. This can be represented by the equation l = w + 12, where l is the length and w is the width.
Q: What is the perimeter of the basketball field?
A: The perimeter of the basketball field is given as 96 meters.
Q: How can we use the perimeter to find the dimensions of the field?
A: We can use the formula for the perimeter of a rectangle, P = 2l + 2w, and substitute the given values to solve for the dimensions of the field.
Q: What is the first step in solving the problem?
A: The first step is to substitute the equation l = w + 12 into the perimeter equation P = 2l + 2w.
Q: How do we simplify the equation?
A: We can simplify the equation by expanding and combining like terms.
Q: What is the final equation we get after simplifying?
A: The final equation is 96 = 4w + 24.
Q: How do we solve for the width?
A: We can solve for the width by subtracting 24 from both sides of the equation and then dividing both sides by 4.
Q: What is the value of the width?
A: The value of the width is 18 meters.
Q: How do we find the length?
A: We can find the length by substituting the value of the width into the equation l = w + 12.
Q: What is the value of the length?
A: The value of the length is 30 meters.
Q: What are the dimensions of the basketball field?
A: The dimensions of the basketball field are 18 meters wide and 30 meters long.
Q: What is the main concept used to solve this problem?
A: The main concept used to solve this problem is algebraic equations, specifically substitution and elimination methods.
Q: What are some real-world applications of this problem?
A: This problem has real-world applications in fields such as architecture, engineering, and construction.
Q: What are some common mistakes to avoid when solving this problem?
A: Some common mistakes to avoid when solving this problem include failing to follow the order of operations (PEMDAS), not using substitution and elimination methods, and not checking the solution to make sure it is reasonable.
Q: What are some future directions for this problem?
A: Some future directions for this problem include extending it to more complex problems, such as finding the dimensions of a field with a non-rectangular shape, and exploring more advanced topics in mathematics, such as calculus and differential equations.
Conclusion
In this Q&A article, we have discussed the problem of finding the dimensions of a basketball field given its perimeter and the relationship between its length and width. We have used algebraic equations, specifically substitution and elimination methods, to solve the problem and have identified some real-world applications and common mistakes to avoid. We have also discussed some future directions for this problem, including extending it to more complex problems and exploring more advanced topics in mathematics.