The Perimeter Of A Rectangle Is 15x + 17yIf The Length Is X + 7y Then Find The Width Of Therectangle.
Understanding the properties and formulas related to rectangles is fundamental in geometry, especially when dealing with algebraic expressions involving variables. In this article, we will explore how to determine the width of a rectangle when given its perimeter and length expressed in algebraic terms. This problem involves applying the perimeter formula of a rectangle and algebraic manipulation to solve for the unknown width.
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Understanding the Perimeter of a Rectangle
What Is the Perimeter?
The perimeter of a rectangle is the total distance around the figure. It is calculated by summing the lengths of all four sides. In a rectangle, opposite sides are equal in length, which simplifies the perimeter calculation.Perimeter Formula of a Rectangle
The standard formula for the perimeter (P) of a rectangle is:\[ P = 2 \times (\text{length} + \text{width}) \]
where:
- Length (L) is the longer side of the rectangle.
- Width (W) is the shorter side.
When the length and width are expressed algebraically, the perimeter becomes a function of variables, often involving expressions like 15x + 17y.
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Given Data and Problem Statement
The problem provides the following information:
- Perimeter of the rectangle: \( P = 15x + 17y \)
- Length of the rectangle: \( L = x + 7y \)
Our goal:
- Find the width \( W \) of the rectangle in terms of variables x and y.
This problem involves algebraic expressions and requires applying the perimeter formula, substituting the known length, and solving for the width.
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Step-by-Step Solution Approach
1. Recall the Perimeter Formula
\[ P = 2 \times (L + W) \]Given that:
- \( P = 15x + 17y \)
- \( L = x + 7y \)
We can substitute these values into the perimeter formula to find \( W \).
2. Set Up the Equation
\[ 15x + 17y = 2 \times [(x + 7y) + W] \]This is an algebraic equation with the unknown \( W \).
3. Simplify the Equation
Distribute the 2 on the right side: \[ 15x + 17y = 2x + 14y + 2W \]Now, to isolate \( W \), move all known terms to one side:
\[
15x + 17y - 2x - 14y = 2W
\]
Simplify the left side:
\[
(15x - 2x) + (17y - 14y) = 2W
\]
\[
13x + 3y = 2W
\]
4. Solve for the Width \( W \)
Divide both sides of the equation by 2: \[ W = \frac{13x + 3y}{2} \]This expression represents the width of the rectangle in terms of variables \( x \) and \( y \).
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Final Expression for the Width of the Rectangle
\[
\boxed{
W = \frac{13x + 3y}{2}
}
\]
This formula provides a direct way to compute the width given specific values of \( x \) and \( y \).
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Interpreting the Result
The derived expression shows that the width is dependent on the variables \( x \) and \( y \). To find a numerical value for the width, you need specific values of these variables.
Example:
Suppose:
- \( x = 2 \)
- \( y = 3 \)
Substitute into the expression:
\[
W = \frac{13(2) + 3(3)}{2} = \frac{26 + 9}{2} = \frac{35}{2} = 17.5
\]
So, the width of the rectangle would be 17.5 units.
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Additional Considerations and Real-World Applications
Practical Applications of the Formula
Understanding how to manipulate algebraic expressions for geometric figures has numerous real-world applications, including:- Architecture and construction planning.
- Design of rectangles in manufacturing.
- Space optimization in layouts.
Constraints and Validity
- The values of \( x \) and \( y \) should be chosen such that the width \( W \) remains positive.
- The variables \( x \) and \( y \) could represent different parameters depending on context, such as dimensions, scaling factors, or other quantities.
Summary and Key Takeaways
- The perimeter of a rectangle can be expressed algebraically as \( P = 2 (L + W) \).
- Given the perimeter \( 15x + 17y \) and length \( x + 7y \), we can derive the width \( W \) using algebraic substitution.
- The final formula for the width is:
- To find the numerical value of \( W \), specific values of \( x \) and \( y \) are needed.
Conclusion
Understanding the relationship between the perimeter, length, and width of a rectangle is essential in solving geometry problems involving algebraic expressions. By applying the perimeter formula and algebraic manipulation, we obtained a clear formula for the width of the rectangle based on the given parameters. This approach exemplifies the importance of combining geometric concepts with algebraic skills to solve real-world problems efficiently.
Remember, always verify the values of your variables to ensure the dimensions make sense in the context of the problem, and practice applying these methods to a variety of similar problems for mastery.
