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Chapter 3: Problem 13

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

Expert verified
The width of the boardwalk is 5 feet.

Step by step solution

01

- 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.
02

- 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.
03

- 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 \[\text{Area} = (30 + 2x)(40 + 2x) = 2000 \text{ ft}^2\]
04

- 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: \[1200 + 60x + 80x + 4x^2 = 2000.\]
05

- 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: \[4x^2 + 140x - 800 = 0.\]
06

- 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: \[x = \frac{-140 \pm \sqrt{140^2 - 4 \cdot 4 \cdot (-800)}}{2 \cdot 4}\]
07

- 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, \(x = 5\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

The volume of water in a rectangular fish tank can be modelled by the polynomial \(V(x)=x^{3}+14 x^{2}+63 x+90 .\) If the depth of the tank is given by the polynomial \(x+6,\) what polynomials represent the possible length and width of the fish tank?

Populations in rural communities have declined in Western Canada, while populations in larger urban centers have increased. This is partly due to expanding agricultural operations and fewer traditional family farms. demographer uses a polynomial function to predict the population, \(P,\) of a town \(t\) years from now. The function is \(P(t)=t^{4}-20 t^{3}-20 t^{2}+1500 t+15000\) Assume this model can be used for the next 20 years.a) What are the key features of the graph of this function? b) What is the current population of this town? c) What will the population of the town be 10 years from now? d) When will the population of the town be approximately 24 000?

For each function, determine i) the \(x\) -intercepts of the graph ii) the degree and end behaviour of the graph iii) the zeros and their multiplicity iv) the \(y\) -intercept of the graph v) the intervals where the function is positive and the intervals where it is negative a) \(y=x^{3}-4 x^{2}-45 x\) b) \(f(x)=x^{4}-81 x^{2}\) c) \(h(x)=x^{3}+3 x^{2}-x-3\) d) \(k(x)=-x^{4}-2 x^{3}+7 x^{2}+8 x-12\)

a) Divide the polynomial $$3 x^{4}-4 x^{3}-6 x^{2}+17 x-8 \text { by } x+1$$ using long division. Express the result in the form \(\frac{P(x)}{x-a}=Q(x)+\frac{R}{x-a}\) b) Identify any restrictions on the variable. c) Write the corresponding statement that can be used to check the division. d) Verify your answer.

For what value of \(c\) will the polynomial \(P(x)=-2 x^{3}+c x^{2}-5 x+2\) have the same remainder when it is divided by \(x-2\) and by \(x+1 ?\)
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