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Thread: applications of quadratic equations, geometric problem

  1. #1
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    applications of quadratic equations, geometric problem

    The problem: find the dimensions of a rectangle whose area is 180cm2 and whose perimeter is 54cm.

    What I know: area = length*width. If the perimeter is 54cm, then would the length * width be 1/2 the perimeter?

    I am also unsure about the dimensions because I do not know how to represent 180cm2 in the equation. I tried to use a metric converter to feet and inches, but I had difficulty converting CM2 to feet and inches. Not that the dimensions need to be converted, I think all the calculations can be done in centimeters, but I am confused.

    Please help me formulate the equation
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  2. #2
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    Re: applications of quadratic equations, geometric problem

    $A = \ell \cdot w = 180$

    $P = 2(\ell + w) = 54$

    $\ell + w = 27$

    $\ell = 27-w$

    $A = (27-w)w = 27w-w^2 = 180$

    $w^2 - 27w + 180 = 0$

    $(w-12)(w-15) = 0$

    $w = 12 \text{ or }w=15$

    $w=12 \Rightarrow \ell = 27-12 = 15$

    $w=15 \Rightarrow \ell = 27-15 = 12$

    so the dimensions of the rectangle are $15cm \times 12cm$ in either horizontal or vertical orientation.
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  3. #3
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    Re: applications of quadratic equations, geometric problem

    Quote Originally Posted by retro View Post
    The problem: find the dimensions of a rectangle whose area is 180cm2 and whose perimeter is 54cm.

    What I know: area = length*width. If the perimeter is 54cm, then would the length * width be 1/2 the perimeter?
    No, why would you even think that? "Perimeter" is the total distance around the rectangle, "length plus width plus length plus width"= 2*length+ 2*width= 2(length+ width). Half the perimeter is length plus width, NOT "length times width".

    I am also unsure about the dimensions because I do not know how to represent 180cm2 in the equation. I tried to use a metric converter to feet and inches, but I had difficulty converting CM2 to feet and inches. Not that the dimensions need to be converted, I think all the calculations can be done in centimeters, but I am confused.

    Please help me formulate the equation
    Why would you want to convert? You are given the area in square cm and perimeter in cm. You are asked for width and length. Just calculate them in centimeters.
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    Re: applications of quadratic equations, geometric problem

    Thank you
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