Question

A boardwalk that is \(x\) feet wide is built around a rectangular pond. The pond is 30 ft wide and 40 ft long. The combined surface area of the pond and the boardwalk is \(2000 \mathrm{ft}^{2} .\) What is the width of the boardwalk?

Short answer

The width of the boardwalk is 5 feet.

Step-by-step solution

- Define Variables

Let the width of the boardwalk be denoted by the variable \(x\) feet. We know the dimensions of the pond are 30 feet by 40 feet.

- Calculate the Dimensions of the Entire Area

The dimensions of the combined surface including both the pond and the boardwalk will be increased by the width of the boardwalk on each side. This means adding \(2x\) to each dimension of the pond: Width: \(30 + 2x\) feetLength: \(40 + 2x\) feet.

- Set Up the Area Equation

The combined area of the pond and the boardwalk is given as 2000 square feet. Using the dimensions calculated in the previous step, the combined area can be expressed as \[\text{Area} = (30 + 2x)(40 + 2x) = 2000 \text{ ft}^2\]

- Expand the Equation

Expand the equation \[(30 + 2x)(40 + 2x)\]Using the distributive property, we get: \[30 \times 40 + 30 \times 2x + 40 \times 2x + 2x \times 2x = 2000\] This simplifies to: \[1200 + 60x + 80x + 4x^2 = 2000.\]

- Simplify the Equation

Combine like terms and write in the form of a quadratic equation: \[4x^2 + 140x + 1200 = 2000.\] Subtract 2000 from both sides: \[4x^2 + 140x - 800 = 0.\]

- Solve the Quadratic Equation

To find the value of \(x\), use the quadratic formula: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] In this equation, \(a = 4\), \(b = 140\), and \(c = -800\). Plugging these values into the quadratic formula gives: \[x = \frac{-140 \pm \sqrt{140^2 - 4 \cdot 4 \cdot (-800)}}{2 \cdot 4}\]

- Calculate Discriminant and Roots

Calculate the discriminant: \[b^2 - 4ac = 140^2 - 4 \cdot 4 \cdot (-800) = 19600 + 12800 = 32400\]Calculate the roots: \[x = \frac{-140 \pm \sqrt{32400}}{8} = \frac{-140 \pm 180}{8}\] The solutions are: \[x = \frac{40}{8} = 5\] and \[x = \frac{-320}{8} = -40.\] Since \(x\) represents a width, it cannot be negative. Therefore, \(x = 5\).

Key concepts

quadratic equations

Quadratic equations are mathematical expressions of the form \(ax^2 + bx + c = 0\). In this problem, we derived a quadratic equation when setting up the expression for the combined area of the pond and the boardwalk. The equation was derived from \((30 + 2x)(40 + 2x) = 2000\), expanding and simplifying gave us \(4x^2 + 140x - 800 = 0\) where:

• \(a = 4\)
  
• \(b = 140\)
  
• \(c = -800\)
  

To solve it, we used the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Understanding the discriminant \(b^2 - 4ac\) is crucial, as it helps determine the nature of the roots.Here, we found that the discriminant is 32400, indicating two real and distinct roots. This step leads us to find the width of the boardwalk, which is the positive root value.

surface area

Surface area refers to the total area covered by a 2-dimensional shape. In the exercise, the combined surface area of the pond and the boardwalk was given as 2000 square feet.
The dimensions of the pond are 30 feet by 40 feet. When adding the boardwalk, the new dimensions become \(30 + 2x\) and \(40 + 2x\).
The total surface area (A) can be found using the area formula for rectangles:

• \(A = \text{length} \times \text{width}\)

Thus, the entire surface area combining the pond and the boardwalk is \((30 + 2x)\times (40 + 2x) = 2000 \text{ft}^2\).
This exercise highlights understanding how dimensions affect the overall surface area and the importance of setting up and solving the correct area equations to find the unknown variable, x.

algebraic manipulation

Algebraic manipulation involves rearranging and simplifying algebraic expressions to solve equations. In the given exercise, various algebraic techniques were used:

• Expanding the Expressions: We expanded the expression \((30 + 2x)(40 + 2x)\) to get \(1200 + 60x + 80x + 4x^2\).
• Combining Like Terms: The terms \(60x\) and \(80x\) were combined to simplify the equation to \(4x^2 + 140x + 1200 = 2000\).
• Isolating the Variable: We subtracted 2000 from both sides to set the equation to \(4x^2 + 140x - 800 = 0\).
• Using the Quadratic Formula: Finally, we applied the quadratic formula to find the values of \(x\).

Mastering algebraic manipulation is essential for solving complex problems, as it allows for simplifying and finding unknown values effectively.

problem-solving

Effective problem-solving requires breaking down the problem into manageable steps, as shown in the exercise. The steps were:

• Step 1: Define Variables
  Identify the unknown (the width of the boardwalk) and represent it with a variable (\(x\)).
• Step 2: Calculate Dimensions
  Determine the new dimensions by adding the width of the boardwalk to each side.
• Step 3: Set Up the Area Equation
  Use given conditions to set up an equation. Here, it was for the combined area.
• Step 4: Expand and Simplify
  Expand the equation using algebra and simplify it into a standard quadratic form (\(ax^2 + bx + c = 0\)).
• Step 5: Solve the Equation
  Use the quadratic formula to find the variable's value (\(x\)).
• Step 6: Interpret Results
  Check the solutions to ensure they fit the context of the problem, choosing the realistic positive value for the width.

Each step builds on the previous one, highlighting the importance of a systematic approach to solving math problems.