Math Problem Statement
Solution
The problem presented in your screenshot states:
- A rectangular garden is 30 ft longer than it is wide.
- The area of the garden is 5,400 square feet.
We need to find the width and length of the garden.
Step-by-step solution:
Let the width of the garden be feet.
Since the length is 30 feet longer than the width, the length will be feet.
The area of a rectangle is given by: Substituting the given values:
Expanding the equation:
Rearranging the equation into standard quadratic form:
Solving the quadratic equation:
We can solve this quadratic equation using the quadratic formula: For the equation , the coefficients are:
Substitute these into the quadratic formula and solve for .
Let me calculate this for you.The solutions to the quadratic equation are:
- (positive value)
- (negative value, which doesn't make sense in this context since width can't be negative).
Therefore, the width of the garden is feet.
Length:
The length is 30 feet longer than the width:
Final dimensions:
- Width = 60 feet
- Length = 90 feet
Would you like further clarification on any steps?
Here are some related questions you might be interested in:
- How do you complete the square to solve quadratic equations?
- What are other methods to solve quadratic equations?
- How can you graphically solve a quadratic equation?
- What are real-world applications of quadratic equations in geometry?
- How do you derive the quadratic formula?
Tip: When solving a quadratic equation, always check both solutions, but discard any that don't make sense in the context (e.g., negative lengths in geometry).
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Math Problem Analysis
Mathematical Concepts
Geometry
Quadratic Equations
Algebra
Formulas
Area of rectangle: Area = length × width
Quadratic formula: x = (-b ± √(b² - 4ac)) / 2a
Theorems
Quadratic formula
Suitable Grade Level
Grades 9-10