A rectangle has an area of 40 square units. The length is 6 units greater than the width. What are the dimensions of the rectangle?
A. 8 by 5
B. 10 by 4
C. 11 by 9
D. 13 by 7
Answer (1)
Define the width as w and express the length as l = w + 6 .
Formulate the area equation: w ( w + 6 ) = 40 .
Solve the quadratic equation w 2 + 6 w − 40 = 0 by factoring to find w = 4 (since width must be positive).
Calculate the length: l = 4 + 6 = 10 . The dimensions are 10 by 4 .
Explanation
Problem Analysis Let's analyze the problem. We are given that the area of a rectangle is 40 square units, and the length is 6 units greater than the width. We need to find the dimensions (length and width) of the rectangle.
Define Variables Let's define our variables. Let w represent the width of the rectangle, and let l represent the length of the rectangle. We know that l = w + 6 and the area A = l × w = 40 .
Substitute Now, we can substitute the expression for l into the area equation: 40 = ( w + 6 ) × w .
Expand Expanding the equation, we get 40 = w 2 + 6 w .
Rearrange Rearranging the equation into a quadratic equation, we have w 2 + 6 w − 40 = 0 .
Factor We can solve this quadratic equation by factoring. We are looking for two numbers that multiply to -40 and add to 6. These numbers are 10 and -4. So, we can factor the quadratic equation as ( w + 10 ) ( w − 4 ) = 0 .
Solve for w This gives us two possible solutions for w : w = − 10 or w = 4 . Since the width must be positive, we choose the positive solution: w = 4 .
Solve for l Now, we can calculate the length using the equation l = w + 6 . Substituting w = 4 , we get l = 4 + 6 = 10 .
Final Dimensions Therefore, the dimensions of the rectangle are 10 units by 4 units.
Examples
Understanding the dimensions of rectangles is crucial in various real-life scenarios. For instance, when designing a garden, knowing the area and the relationship between length and width helps in planning the layout efficiently. If you have a fixed area and want to optimize the perimeter (e.g., to minimize fencing), this type of problem becomes essential. Similarly, in architecture and interior design, determining dimensions based on area constraints is a common task. Let's say you have a rectangular room with an area of 40 square meters, and you want the length to be 6 meters longer than the width. By solving the quadratic equation, you can find the exact dimensions needed for your design.
