Question

A machine shop is manufacturing a pair of gears that need to be in a ratio as close to 1.1839323 as possible, but they can’t make gears with more than 50 teeth on them. How many teeth should be on each gear to best approximate this ratio? (Reminder: what concept from this course is relevant to this problem?)

Answer 1

Answer:

45 and 38

Explanation:

Note that ratio of 2 gears are N1/N2 such that N1 is the first gear, N2 is the second gear

N1/N2 = 1.1839323 =$\frac{11839323}{10000000}$

          = $1\frac{11839323}{10000000}$

Express this as continued function, we will get as follows

1; 5, 2, 3, 2, 5, .....

Using this continued function, try find the least error while trying to find the combination of nominator and denominator with both less than 50

[1;] = 1                         error -0.1839323        1/1

[1;3] = 1.33...                 error +0.1494010       4/3

[1;4] = 1.25                 error +0.0660677       5/4

[1;5] = 1.2                         error +0.0160677        6/5

[1;5,2] = 1.18182               error -0.002114118      13/11

[1;5,2,2] = 1.185185 error +0.001253          32/27

[1;5,2,3] = 1.184210 error +0.000278         45/38

[1;5,2,3,1] = 1.183673 error -0.0002588        58/49

[1;5,2,3,2] = 1.18391 error -0.000024254    103/87

at  [1;5,2,3,1], the value of  nominator and denominator has exceeded the limit 50,

So the nearest ratio suitable for this problem is 1.184210 with ratio 45/38 i.e. gear 1 with 45 teeth, gear 2 with 38 gear.