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Math Help - Problem solving with Quadratics x2

  1. #1
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    Exclamation Problem solving with Quadratics x2

    Find the width of a uniform concrete path placed around a 30 m by 40 m rectangular lawn given that the concrete has area one quarter of the lawn.
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  2. #2
    Super Member Bacterius's Avatar
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    Hello,
    Say the area of the lawn is A, and the area of the path is A'. We are given the dimensions of the lawn, so A = 30 \times 40 = 1200.

    Now you know that A' = \frac{A}{4} = 300.

    Let us define a function that gives the area of the concrete path with respect to some width, denoted x. After investigation, we find that this function is defined by : f(x) = 60x + 2(40 - 2x)x = 60x + (80 - 4x)x = 60x + 80x - 4x^2 = -4x^2 + 140x. You are trying to find x such as f(x) = 300. Solve -4x^2 + 140x = 300 for x :

    -4x^2 + 140x - 300 = 0

    Using the quadratic formula :

    \Delta = 140^2 - 4 \times (-4) \times (-300) = 14800

    x_1 = \frac{-140 - \sqrt{14800}}{-8}

    x_2 = \frac{-140 + \sqrt{14800}}{-8}

    A width can only be positive, so discard the negative solution. You just found the width of the concrete path

    Does that make sense ?
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  3. #3
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    Quote Originally Posted by Bacterius View Post
    Hello,
    Say the area of the lawn is A, and the area of the path is A'. We are given the dimensions of the lawn, so A = 30 \times 40 = 1200.

    Now you know that A' = \frac{A}{4} = 300.

    Let us define a function that gives the area of the concrete path with respect to some width, denoted x. After investigation, we find that this function is defined by : f(x) = 60x + 2(40 - 2x)x = 60x + (80 - 4x)x = 60x + 80x - 4x^2 = -4x^2 + 140x. You are trying to find x such as f(x) = 300. Solve -4x^2 + 140x = 300 for x :

    -4x^2 + 140x - 300 = 0

    Using the quadratic formula :

    \Delta = 140^2 - 4 \times (-4) \times (-300) = 14800

    x_1 = \frac{-140 - \sqrt{14800}}{-8}

    x_2 = \frac{-140 + \sqrt{14800}}{-8}

    A width can only be positive, so discard the negative solution. You just found the width of the concrete path

    Does that make sense ?
    Thank you for helping.

    I tried the answer you gave me but it doesn't match up with the answer in the back of my math book. The answer i have in my math text book is: 2.026 m.
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