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Thread: 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 $\displaystyle A$, and the area of the path is $\displaystyle A'$. We are given the dimensions of the lawn, so $\displaystyle A = 30 \times 40 = 1200$.

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

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

    $\displaystyle -4x^2 + 140x - 300 = 0$

    Using the quadratic formula :

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

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

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

    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 $\displaystyle A$, and the area of the path is $\displaystyle A'$. We are given the dimensions of the lawn, so $\displaystyle A = 30 \times 40 = 1200$.

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

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

    $\displaystyle -4x^2 + 140x - 300 = 0$

    Using the quadratic formula :

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

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

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

    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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