Math Problem Statement
The area of a rectangle is52yd^2, and the length of the rectangle is 5yd less than twice the width. Find the dimensions of the rectangle.
Solution
Let the width of the rectangle be yards.
Then, the length of the rectangle is given as 5 yards less than twice the width, which can be expressed as:
The area of the rectangle is given as 52 square yards, so we use the formula for the area of a rectangle:
Substituting the given values:
Step 1: Expand the equation
Step 2: Rearrange into standard quadratic form
Step 3: Solve for using the quadratic formula
The quadratic formula is:
For the equation , we have:
First, compute the discriminant:
Now, take the square root:
Now, solve for :
Step 4: Find possible values for
Since the width cannot be negative, we take:
Step 5: Find the length
Final Answer:
- Width = 6.5 yards
- Length = 8 yards
Would you like a more detailed breakdown or verification of the steps?
Related Questions:
- How do you solve quadratic equations by factoring instead of using the quadratic formula?
- What are the properties of a rectangle that distinguish it from other quadrilaterals?
- How can you determine the dimensions of a rectangle if given perimeter instead of area?
- How does changing the width affect the area of a rectangle while keeping the length proportional?
- What real-world applications involve finding dimensions of a rectangle given area constraints?
Math Tip:
Always check whether the quadratic equation can be factored easily before using the quadratic formula, as factoring can save time. 🚀
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Algebra
Quadratic Equations
Geometry
Formulas
Area of a rectangle = Length × Width
Quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a
Theorems
Quadratic formula
Suitable Grade Level
Grades 7-9
Related Recommendation
Find Dimensions of a Rectangle: Algebraic and Geometric Solution
Find Rectangle Dimensions Using Ratio of Length to Width and Area
Dimensions of a Rectangular Window: Solving for Length and Width
Solve the Problem of Rectangle Dimensions from Area Calculation
Finding the Width of a Rectangle Using Quadratic Factoring