In an orthogonal machining with a tool of $${9^ \circ }$$ orthogonal rake angle, the uncut chip thickness is $$0.2$$ $$mm.$$ The chip thickness fluctuates between $$0.25$$ $$mm$$ and $$0.4$$ $$mm.$$ The ratio of the maximum shear angle to the minimum shear angle during machining is ______________

Two cutting tools with tool life equations given below are being compared:
$$\eqalign{ & Tool\,\,1:\,\,\,\,\,\,\,V{T^{0.1}} = 150 \cr & Tool\,\,2:\,\,\,\,\,\,\,V{T^{0.3}} = 300 \cr} $$
where $$V$$ is cutting speed in $$m/minute$$ and $$T$$ is tool life in minutes. The breakeven cutting speed beyond which Tool $$2$$ will have a higher tool life is _______________ $$m/minute$$.

For a certain job, the cost of metal cutting is $$Rs.18C/V$$ and the cost of tooling is Rs.$$270C/(TV).$$ Where $$C$$ is a constant, $$V$$ is the cutting speed in $$m/min$$ and $$T$$ is the tool life in minutes. The Taylor's tool life equation is $$V{T^{0.25}} = 150.$$ The cutting speed (in $$m/min$$) for the minimum total cost is ___________

In a single point turning operation with cemented carbide tool and steel work piece, it is found that the Taylor's exponent is $$0.25.$$ If the cutting speed is reduced by $$50\% $$ then the tool life changes by ____________ times.