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Math Help - Surds

  1. #1
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    Surds

    Please help....

    A large rectangular piece of card is square root of 5 + square root of 20 cm long and square root of 8 cm wide. A small rectangle square root of 2 cm long and square root of 5 cm wide is cut out of the piece of card. Express the area of the card that is left in a percentage of the area of the large rectangle?
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  2. #2
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    e^(i*pi)'s Avatar
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    Let A_1 be the large piece and A_2 the small piece

    The ratio of the card left is therefore given by \dfrac{A_1-A_2}{A_1} \times 100\%

    A_1 = (\sqrt{5} + \sqrt{20}) \times \sqrt{8}

    Since \sqrt{20} = \sqrt{4\cdot 5} = 2\sqrt{5} and that \sqrt{8} = \sqrt{2^2 \cdot 2} = 2\sqrt{2} we can rewrite the above equation:

    A_1 = (\sqrt{5}+2\sqrt{5}) \ times 2\sqrt{2} = 3\sqrt{5} \times 2\sqrt{2} = 6\sqrt{10}

    Note I have used the rule of surds which says that \sqrt{a} \cdot \sqrt{b} = \sqrt{ab}\: \: a,b \geq 0


    Can you find A_2 using the same principles?
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  3. #3
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    A2 = square root of 2 x square root of 5 = Square root of 10

    How do we express the area of the card that is left as a percentage of the area of the big rectangle?

    Square root of 10 divided by 6 Square root of 10???
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  4. #4
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    Quote Originally Posted by Natasha1 View Post
    A2 = square root of 2 x square root of 5 = Square root of 10

    How do we express the area of the card that is left as a percentage of the area of the big rectangle?

    Square root of 10 divided by 6 Square root of 10???
    No, " \sqrt{10}" is NOT the "area of the card that is left". It is the area that is removed.
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  5. #5
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    Is the answer 83.3 percent?
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  6. #6
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    Quote Originally Posted by Natasha1 View Post
    Is the answer 83.3 percent?

    the answer looks right.
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  7. #7
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    Quote Originally Posted by Natasha1 View Post
    Is the answer 83.3 percent?
    \dfrac{6\sqrt{10} - \sqrt{10}}{6\sqrt{10}} = \dfrac{5}{6} \approx 83.3\%
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