Math Problem Statement

A rectangular garden is 30 ft longer than it is wide. Its area is 5,400 ft². What are its dimensions (in ft)?

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 ww feet.

Since the length is 30 feet longer than the width, the length will be w+30w + 30 feet.

The area of a rectangle is given by: Area=length×width\text{Area} = \text{length} \times \text{width} Substituting the given values: 5400=w×(w+30)5400 = w \times (w + 30)

Expanding the equation: 5400=w2+30w5400 = w^2 + 30w

Rearranging the equation into standard quadratic form: w2+30w5400=0w^2 + 30w - 5400 = 0

Solving the quadratic equation:

We can solve this quadratic equation using the quadratic formula: w=b±b24ac2aw = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} For the equation w2+30w5400=0w^2 + 30w - 5400 = 0, the coefficients are:

  • a=1a = 1
  • b=30b = 30
  • c=5400c = -5400

Substitute these into the quadratic formula and solve for ww.

Let me calculate this for you.The solutions to the quadratic equation are:

  • w1=60w_1 = 60 (positive value)
  • w2=90w_2 = -90 (negative value, which doesn't make sense in this context since width can't be negative).

Therefore, the width of the garden is 6060 feet.

Length:

The length is 30 feet longer than the width: Length=60+30=90feet\text{Length} = 60 + 30 = 90 \, \text{feet}

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:

  1. How do you complete the square to solve quadratic equations?
  2. What are other methods to solve quadratic equations?
  3. How can you graphically solve a quadratic equation?
  4. What are real-world applications of quadratic equations in geometry?
  5. 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