1 A roll of crackers is approximately 360mm diameter. Each cracker is approximately 6mm diameter. The roll is a loose spiral. Show that r=6Ɵ/π - 3 is an appropriate form for the line of centres and find the range of Ɵ. [4]
ii Find the length of a roll in terms of π and count the crackers in this roll. [4]
iii The longest I timed a roll of crackers firing was a minute. How many shots per second does this imply? [2]
Use your own knowledge and observations to criticise my model and suggested figures [3]
2 A rocket explodes level with the 30th floor. If a storey height of between 2.5 and 3 metres is assumed, what height, H, does this suggest? [1]
ii If we model the rocket as a projectile which reaches its maximum height at H, and if we also ignore air resistance, what was the minimum initial velocity, U? Express U exactly for H=98. [3]
iii If we allow for air resistance in the form mkv, develop an equation for s in terms of v on the assumption of vertical travel.
Iv Declare an upper limit for k in terms of g and U. Find U for the occasion when k has the value g/100. [5]
3 Surgical needles are measured in integers by a (slightly odd, but genuine) unit, the gauge [pronounced ‘gayj’]. A similar unit is the ‘french’]. The table below shows some outside diameters of a given gauge, G.
G 10 14 18 20 23 27 30 33
X 3.404 2.108 1.270 0.902 0.635 0.406 0.305 0.203
i Demonstrate that this is not modelled by a linear function. [3]
ii Use logarithmic regression based on the assumption that y = peqx, finding p,q (and r). Express this in the form Y= axb, giving a and b to 3 sig.fig. [6]
iii Find the first integer gauge of a needle whose diameter is less than 0.1mm. Comment on your result. [3]
For a writing mark, give at least one reason or excuse why needles apply at New Year [3]
iv Use a chi squared test to give a measure of confidence in your answer to (ii). [6]
4 Several of you claimed in your last questionnaire that I do not set a written homework each week. Therefore, please
i Explain why the cycle of dates for the Chinese New Year repeats every 60 years [3]
ii Explain why, when it allegedly started in 2637BC, this new year is numbered 4706, not 4645. [3]
iii The cycle of years is a combination of solar and lunar cycles. 13 lunar cycles gives 354±1 days in a year. Explain why this means that there are 7 leap months every 19 years. [3]
iv Apparently the leap month could (or according to some, should) have occurred after the 1st month in 1651 (it didn’t) and the next such occasion would then be in 2262. Explain the interval and predict the next one. [3]
5 The Moon is a spherical body. Therefore its inertia should be 2/5 Mr². Work this through from first principles as follows:
I= z²dm, but z =r sin, dm = dV and dV should be r² sin d d . Write this as a triple integral and establish the limits. [4]
Integrate the r term; integrate for (show enough, here). [3]
This should leave just d to integrate. Apply the obvious = M/V, substitute for V and check your answer: any mistakes probably lie in your arrangement of limits. [2]
6 Apparently the Moon’s effect on the Earth includes the fact that the line of the bulge made by the tides is a little in advance of the Moon itself. This results in the Earth’s rotation becoming a little slower, in the region of 0.7 seconds per year. The last ‘leap second’ added was 1st Jan 2006. The second is defined as 1/86400 of the mean solar year.
i) Using the 0.7 given to 1sf, estimate how many secs have been added since 1972 [1]
X ‘72 ‘80 ‘84 ‘90 ‘94 ‘98 ‘08
T -10 -21.5 -22 -24 -27.5 -30.5 -33.5
ii) The figures show something of the change from 1972. Comment on the linearity using a suitable test; identify the T value with the biggest error and predict the next leap second - with explanation. [4,2,2]
I will happily accept better suggestions for the curve of centres in Q1. It needs to be an Archimedean spiral and, if my geometry was better, I’d see how to construct the required curve.