Math Problem Statement
A person has a garden that has a length 4 feet longer than the width. Set up a quadratic equation to find the dimensions of the garden if its area is 60
ft^2. Solve the quadratic equation to find the length and width.
Enter the exact answers.
Solution
Let's break down the information given:
- Let the width of the garden be feet.
- Since the length is 4 feet longer than the width, the length will be feet.
- The area of the garden is given as 60 square feet.
Using the formula for the area of a rectangle, which is:
we can set up the equation:
Step 1: Set up the quadratic equation
Expanding the equation:
Then, rearrange it to bring all terms to one side:
Step 2: Solve the quadratic equation
Now we have a quadratic equation:
To solve for , we can use the quadratic formula:
where , , and .
Substitute values:
Calculate the two possible values of :
- (We discard this negative value because dimensions cannot be negative.)
So, the width is 6 feet.
Step 3: Find the length
Since the length is :
Final Answer:
The dimensions of the garden are:
- Width: 6 feet
- Length: 10 feet
Would you like more details or have any questions?
Here are five related questions to deepen your understanding:
- What would the quadratic equation be if the area was given as 80 square feet?
- How would the solution change if the length was 6 feet longer than the width?
- How would you solve this problem using factoring instead of the quadratic formula?
- What are the steps to derive the quadratic formula?
- How does the discriminant in the quadratic formula tell us about possible solutions?
Tip: When setting up quadratic equations for area problems, carefully identify which variable represents width and length to avoid misinterpretation.
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Math Problem Analysis
Mathematical Concepts
Algebra
Quadratic Equations
Geometry
Formulas
Quadratic equation formula x^2 + bx + c = 0
Area of a rectangle (Area = length x width)
Theorems
Quadratic Formula
Suitable Grade Level
Grades 8-10
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