Questions: Polynomial and Rational Functions Word problem involving optimizing area by using a quadratic function Linda has 440 meters of fencing and wishes to form three sides of a rectangular field. The fourth side borders a river and will not need fencing. As shown below, one of the sides has length x (in meters). (a) Find a function that gives the area A(x) of the field (in square meters) in terms of x. A(x) = (b) What side length x gives the maximum area that the field can have? Side length x: meters (c) What is the maximum area that the field can have? Maximum area: square meters

Polynomial and Rational Functions Word problem involving optimizing area by using a quadratic function Linda has 440 meters of fencing and wishes to form three sides of a rectangular field. The fourth side borders a river and will not need fencing. As shown below, one of the sides has length x (in meters). (a) Find a function that gives the area A(x) of the field (in square meters) in terms of x. A(x) = (b) What side length x gives the maximum area that the field can have? Side length x: meters (c) What is the maximum area that the field can have? Maximum area: square meters
Transcript text: Polynomial and Rational Functions Word problem involving optimizing area by using a quadratic function Linda has 440 meters of fencing and wishes to form three sides of a rectangular field. The fourth side borders a river and will not $n$ As shown below, one of the sides has length $x$ (in meters). (a) Find a function that gives the area $A(x)$ of the field (in square meters) in terms of $x$. \[ A(x)=\square \] (b) What side length $x$ gives the maximum area that the field can have? Side length $x: \square$ meters (c) What is the maximum area that the field can have? Maximum area: $\square$ square meters
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Solution

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Solution Steps

Step 1: Define the variables and the problem

Linda has 440 meters of fencing to form three sides of a rectangular field. One side of the rectangle borders a river and does not need fencing. Let \( x \) be the length of the side perpendicular to the river, and \( y \) be the length of the side parallel to the river.

Step 2: Express the perimeter constraint

The total length of the fencing used is 440 meters. Since the fencing is used for three sides: \[ 2x + y = 440 \]

Step 3: Express \( y \) in terms of \( x \)

Solve the perimeter constraint for \( y \): \[ y = 440 - 2x \]

Step 4: Define the area function

The area \( A \) of the rectangle is given by: \[ A = x \cdot y \] Substitute \( y \) from the previous step: \[ A = x(440 - 2x) \] \[ A = 440x - 2x^2 \]

Step 5: Find the value of \( x \) that maximizes the area

To find the maximum area, take the derivative of \( A \) with respect to \( x \) and set it to zero: \[ \frac{dA}{dx} = 440 - 4x \] Set the derivative equal to zero: \[ 440 - 4x = 0 \] \[ 4x = 440 \] \[ x = 110 \]

Step 6: Calculate the maximum area

Substitute \( x = 110 \) back into the area function: \[ y = 440 - 2(110) = 220 \] \[ A = 110 \cdot 220 = 24200 \]

Final Answer

  • (a) The function that gives the area \( A(x) \) of the field is: \[ A(x) = 440x - 2x^2 \]
  • (b) The side length \( x \) that gives the maximum area is: \[ x = 110 \text{ meters} \]
  • (c) The maximum area that the field can have is: \[ \text{Maximum area} = 24200 \text{ square meters} \]
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