If reminiscence serves, an LP is 12 inches in diameter, whereas a CD is 12 centimetres in diameter, so the ratio of the diameters can be 2.54, and the ratio of the areas can be 6.4516.
These numbers are supposedly precise rationals, so we all know for positive that the ratio of areas shouldn’t be precisely $2pi$ (since that is irrational), nevertheless it’s clearly shut.
The “apparent” strategy is to compute $6.4516 – 4 arctan(1)$ (utilizing its Taylor collection) to sufficient precision to know its signal, although that looks like dishonest, because it includes computing $frac{pi}{2}$ as an intermediate step.
An strategy that does not contain realizing $pi$ (or calculating some approximation of it) is as an alternative to compute $sin(6.4516_{radians})$ by hand utilizing the Taylor collection, and see whether or not that produces a optimistic or unfavourable reply. Or equivalently, compute $-cos(1.6129)$ and once more, see if that is optimistic or unfavourable. However these do not seem to be they’re within the spirit of this problem, as computing Taylor collection by hand to 3-digit precision can be extraordinarily tedious.
So I’ll assume that “not use the decimal illustration of $pi$” means we’re allowed to know another approximation
corresponding to $pi approx frac{22}{7}$ (and know that that approximation is excessive), during which case: $6.4516 – 2 × frac{22}{7}$ being optimistic would show that the ratio of areas is greater than $2pi$, however being unfavourable wouldn’t instantly show the it is lower than $2pi$.
So I wish to compute the signal of $(frac{254}{100})^2 – frac{44}{7} = frac{16129}{2500} – frac{44}{7}$
Which is identical because the signal of $16129 × 7 – 44 × 2500$
Which is +2903, so sure the ratio of areas exceeds $2pi$.
