The discussion focuses on calculating the number of smaller circles with a radius of 7.4 m that can fit inside a larger circle with a radius of 50 m, while remaining tangent to its surface without using trigonometric functions. The primary method discussed involves using a Taylor expansion to approximate the relationship, leading to the conclusion that a maximum of 18 smaller circles can fit within the larger circle. Alternative methods, including the use of concentric circles and polygonal approximations, are also explored, emphasizing the importance of accuracy in calculations.
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Thanks for the great work... but is that the closest that we can get ?.. i mean it's the right answer because N is equal 18 but what if in another case N has a fraction ... then by this method we can never know it's accurate valueanuttarasammyak said:A prescription is : Draw two lines from the center of large circle of radius R so that they are tangible to the small circle of radius r. How much is the angle between the lines?
With trigonometric, it is 2\ sin^{-1}\ \frac{r}{R-r}
Similar but another approach is, thinking N-regular polygon inside and tangent to the large circle, find N that satisfies
sin \frac{\pi}{N+1}< \frac{r}{R-r} < sin \frac{\pi}{N}.
Mostly we can replace the most LHS and RHS by their first term of Taylor expansion
\frac{\pi}{N+1}< \frac{r}{R-r} < \frac{\pi}{N}
where no trigonometric appear. For OP's case the middle term is 0.1737... so
N<18.08...<N+1
So N=18. I hope Taylor expansion approximation works at least for the OP's case.
anuttarasammyak said:N is a number of sides of polygon, so is an integer, with no fraction. If you are keen on accuracy, estimate second and higher terms of Taylor expansion or, as the straightest way , use trigonometry.