Math Problem Statement
Solution
The given diagram shows a rectangle with dimensions (length) and (width). The area of the rectangle is 72 cm². We can set up an equation based on the formula for the area of a rectangle, which is:
Substituting the given values:
Expanding the right-hand side:
Now, we have the quadratic equation:
We can solve this quadratic equation using the quadratic formula. The quadratic formula is:
For the equation , we have:
- ,
- ,
- .
Now, let's calculate the discriminant:
Now, applying the quadratic formula:
This gives two possible solutions for :
Since represents a dimension, we discard the negative value. Thus, .
Now, substituting back into the expressions for the dimensions:
- Length:
- Width:
Thus, the dimensions of the rectangle are approximately .
Would you like further details on this? Here are some questions to expand your understanding:
- What would the area be if were a different value?
- Can you solve the quadratic equation using factoring instead of the quadratic formula?
- How would the dimensions change if the area were doubled?
- How do you find the perimeter of this rectangle?
- What are the possible values for the width and length if we allow negative dimensions?
Tip: When solving quadratic equations, always check for negative or non-real solutions, especially when dealing with physical dimensions like length and width.
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Math Problem Analysis
Mathematical Concepts
Algebra
Quadratic Equations
Area of Rectangles
Formulas
Area of a rectangle = length × width
Quadratic equation formula: ax^2 + bx + c = 0
Theorems
Quadratic formula
Suitable Grade Level
Grades 9-10
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