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Math Help - Optimizing fence

  1. #1
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    Optimizing fence

    Thank You in advance to those who try and solve this problem

    Chris needs 50m of fencing to enclose two separate square gardens whose area must differ by no more than 100m squared. What is the greatest length that a side of the larger square garden can have.
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  2. #2
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    Quote Originally Posted by begold91 View Post
    Thank You in advance to those who try and solve this problem

    Chris needs 50m of fencing to enclose two separate square gardens whose area must differ by no more than 100m squared. What is the greatest length that a side of the larger square garden can have.
    Alright, the large garden is garden 1. The small one is garden 2. s_1 means a side of the first garden...

    You know that: s_1^2-s_2^2\leq100

    And that: 4s_1+4s_2=50

    Thus: s_1+s_2=\frac{50}{4}=12.5

    Then: s_2=12.5-s_1

    Now go back to this one: s_1^2-s_2^2\leq100

    Substitute: s_1^2-\left(12.5-s_1\right)^2\leq100

    Then: s_1^2-\left(156.25-25s_1+s_1^2\right)\leq100

    Now spread out the negative sign: s_1^2-156.25+25s_1-s_1^2\leq100

    Group like terms: s_1^2-s_1^2+25s_1-156.25\leq100

    Subtract: 25s_1-156.25\leq100

    Add 156.25 to both sides: 25s_1\leq256.25

    Divide by 25: s_1\leq10.25

    So the greatest length s_1 can have is 10.25. But check my arithmetic!!!
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