Two stocks each swinging 20 percent a year with a correlation of one half give their product a volatility of 20 root 3, or 34.641 percent, rather than 40. Adding the two numbers is correct at exactly one correlation, namely 1, because the composite volatility is the law of cosines with the correlation as the cosine of the angle between two arrows. Pricing the call at 40 percent overstates it by 14.2 percent and ignoring the correlation understates it by 16.9 percent, and the other diagonal of the same parallelogram prices the ratio of the two stocks.
The long hand turns 6 degrees a minute and the short one half a degree, so a 90 degree gap closes at 5.5 degrees a minute and the hands coincide 180/11 minutes past three, at 3:16:21.8181. At 3:15 the long hand has reached the 3 and the short hand is 7.5 degrees ahead of it, which is the whole content of the wrong answer. Consecutive coincidences are 720/11 minutes apart, so there are eleven per twelve hours and twenty-two per day, and the common phrasing "eleven times a day" is wrong by a factor of two.
With no air, v squared equals 2gD gives 89.4 metres per second and t equals root of 2D/g gives 8.94 seconds, both stable under g = 10 or g = 9.81. Dividing the height by the impact speed returns 4.47 seconds, wrong by exactly a factor of two at every drop height, because a body released from rest averages half its final speed. Real air reverses the picture: a coin-sized disc reaches terminal velocity near 11.9 metres per second and takes about 34.6 seconds, so nine seconds is a floor and 200 miles an hour a ceiling.
Running straight out from the centre loses, because a radius costs you 1 while half the fence costs the dog pi over 4. Inside a quarter of the radius your angular speed beats his, so you can orbit until he is diametrically opposite and then sprint three quarters of a radius against his pi over 4, and the whole escape reduces to 3 being less than pi with a margin of 0.0354 R. That two-phase plan works only up to a speed ratio of pi + 1, while the best known strategy for the problem reaches 4.60334.
Differentiating y = L tan(wt) gives a spot speed of w R squared over L, so the footprint accelerates with the square of its distance from the lamp: a tenth of pi directly opposite, and exactly pi miles per second nine miles along. The 9 is the along-shore leg, which makes 90 the squared hypotenuse rather than the square of nine, and that misreading is the usual failure. Nothing physical moves at that speed, and a straight coast running 2310 miles would carry a nominally faster-than-light spot carrying no information at all.
Two riders twenty-five miles apart close at fifty miles an hour, so a forty mile an hour fly shuttling between them flies exactly twenty miles. The series of shuttle legs gives the same twenty, with a first leg of 100/7 and a round-trip ratio of 1/21, but no finite number of legs ever reaches it: after eighty legs the exact total is twenty minus about 2.6e-52. The general law is wD/(a+b), and it holds identically in the rider speeds rather than by luck at 20 and 30.