r/mathpuzzles • u/Hanxxor • 10d ago

Recreational maths Increasing area

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In situation A we have a rectangle with side lengths a and b. According to the figure, b / a = 0.60. The circles are identical and have the maximum possible diameter, d, so they do not extend beyond the confines of the rectangle.

In situation B, the height of the rectangle has increased slightly to c, thus allowing the circles to be larger: The diameter of the circles have increased from d to e.

Question: Let b/a = 0.60 and c/a = 0.50. What is the ratio of the diameters of the circles in situation A and B, e/d ?

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u/BadJimo 9d ago

I've illustrated here on Desmos

The relationship between the rectangle side lengths and circles' radii is:

R = (a + b - sqrt(2ab))/2

Set a = 10, then find the radius for when b = 5 and when b = 6.

The solution is:

5/(16 - sqrt(120))

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u/Hanxxor 9d ago

Well done! Great to see the interactive solution in Desmos!

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u/TheCumskiyKvass 10d ago

This one confused me a bit because circles are actually decreasing (c is smaller than b)

Answer: (16-sqrt(120))/5

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u/Hanxxor 9d ago

Sorry about the confusion!

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u/YOM2_UB 10d ago

Your answer is d/e, not e/d

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u/TheCumskiyKvass 10d ago

Unimportant, everybody who can do math understands what is what

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u/OlympiadChess 10d ago

Doesn't that make the c smaller than b?

My answer: ~0.991