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March 27, 2017

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A shaded circle just fits inside a 2m x 3m rectangle. What is the radius in metres, of the largest circle that will also fit inside the rectangle but will not intersect the shaded circle?

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I have a diagram with me, but I don't think it is possible to upload it, so drawing it out would be a good option.

So, from what we can gather from the question, we know the shaded circle is 2 m in diameter. Many assume the the diameter of the largest circle that can fit into the rectangle would be 1 m. However, if pushed the circle in the corner, there is more space, thus the circle MUST be greater than 1m. How to work out exactly what the diameter is, I have no clue.

Anyone who can comprehend what I ranted about up there and explain how I could solve this question will be hailed the ultimate genius.

Thanks.

  • math - ,

    ----------------------
    I/ \/ ?? \I
    I \ rad I
    I2m diameter I----I
    I circle I I
    I / I
    I\ / I
    ----------------------

    okay... my attempted diagram.. It probably won't show up correctly when i post it, and confuse everyone more... but i thought I'd give it a shot.

  • math - ,

    yup.. it didn't work...
    it deleted all my spacing..
    please ignore the above post.

  • math - ,

    From "A shaded circle just fits inside a 2m x 3m rectangle. What is the radius in metres, of the largest circle that will also fit inside the rectangle but will not intersect the shaded circle?", I assume the circle within the rectangle has a radius of 1 meter.

    Drawing a circle in the space between the circle and opposite end of the rectangle, tangent to the two adjacent sides and the circle has a radius of "r".

    Relative to the given circle and the circle being sought, we can write

    (1 + r) = the distance between the two circle centers
    (2 - r) = the distance between the two centers parallel to the long side of 3m
    The distance between the two centers parallel to the short side of 2m = (1 + r)^2 - (2 - r)^2 = 6r - 3.

    Therefore, r + 6r -3 + 1 = 2 making r = 4/7.

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