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Arithmetic and Geometric progressions?
A drilling machine has ten speeds arranged in GP and operated at a surface cutting speed of 15m/min. If the drill bit capacity is 6-18mm, determine the complete range of speeds.
(speed in m.s^-1=surface cutting speed in m.s^-1/drill bit circumference in m^2)
I have a feeling that it has too much information in order to confuse me, im stuck!
Not too much information, except your "speed in m/s = surface cutting speed..." is wrong. It should have units of "revolutions/minute" instead of meters/second", and the units of circumference should be in meters instead of meters².
It's simpler to calculate if all length dimensions are converted millimeters.
Minimum speed (rpm) = (1500 mm/min)/(18π mm) = 26.5rpm
Maximum speed (rpm) = 1500/6π = 79.5 rpm
So if the drill press has 10 speeds in geometric progression from 26.5 rpm to 79.5 rpm,
S1 = 26.5
S10 = 79.5
Sn+1 = Sn*R where R is the geometric ratio of the progression.
S1 = 26.5
S2 = 26.5R
S3 = S2*R = 26.5R^2
...
S10 = 79.6 = 26.5R^9
R^9 = S10/S1 = (1500/6π)/(1500/18π) = 3
R = 3^(1/9) = 1.1298
Therefore the complete range of speeds is (all rounded to the nearest 0.1 rpm:
S1 = 26.5 rpm
S2 = 29.9 rpm
S3 = 33.8 rpm
S4 = 38.2 rpm
S5 = 43.2 rpm
S6 = 48.8 rpm
S7 = 55.1 rpm
S8 = 62.3 rpm
S9 = 70.4 rpm
S10 = 79.5 rpm
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