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.
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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.
yup.. it didn't work...
it deleted all my spacing..
please ignore the above post.
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.
To solve this problem, we can use the concept of Pythagorean theorem and simple geometry. Let's go step by step:
1. Draw a rectangle with dimensions 2m x 3m on a paper or a digital drawing tool.
2. Draw a circle inside the rectangle that is completely shaded. The circle is said to fit inside the rectangle, so the diameter of the circle should be equal to the shorter side of the rectangle, which is 2m in this case.
3. Now, we need to find the largest circle that can fit inside the rectangle but does not intersect the shaded circle. To do this, we need to find the maximum possible radius for the circle.
4. Place the center of the circle at any of the rectangle's corner. This way, the maximum possible radius will reach as far as possible into the rectangle without intersecting the shaded circle.
5. Now, draw a diagonal from the corner that the circle is placed to the opposite corner of the rectangle.
6. The diagonal of the rectangle divides it into two right-angled triangles. The sides of these triangles are the radius of the largest circle, the width of the rectangle, and the height of the rectangle.
7. Apply the Pythagorean theorem to one of the right-angled triangles. The Pythagorean theorem states that in a right-angled triangle, the sum of the squares of the two shorter sides is equal to the square of the longest side (the hypotenuse).
Let's say the radius of the largest circle is 'r', the width of the rectangle is '2m', and the height of the rectangle is '3m'. The equation using the Pythagorean theorem is:
r^2 + (2m)^2 = (3m)^2
8. Simplify the equation:
r^2 + 4m^2 = 9m^2
9. Subtract 4m^2 from both sides of the equation:
r^2 = 5m^2
10. Take the square root of both sides:
r = sqrt(5m^2)
11. Simplify:
r = m * sqrt(5)
So, the radius of the largest circle that can fit inside the rectangle but does not intersect the shaded circle is 'm * sqrt(5)'.
Now, to find the exact value of the radius in meters, you need to substitute the value of 'm' with the given measurement. In this case, the radius is '2m * sqrt(5)'.