Question

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 VT^0.25 = 150. The cutting speed (in m/min) for the minimum total cost is _______ 

Answer 1

Total cost (C_t)= metal cutting cost + tooling cost

\(\Rightarrow {C_t} = \frac{{18C}}{V} + \frac{{270C}}{{VT}}\)

Also VT^0.25 = 150

\(\Rightarrow T = {\left( {\frac{{150}}{V}} \right)^{\frac{1}{{0.25}}}} = {\left( {\frac{{150}}{V}} \right)^4}\)

\(\therefore {C_t} = \frac{{18C}}{V} + \frac{{270C}}{{V{{\left( {\frac{{150}}{V}} \right)}^4}}}\)

\(\therefore {C_t} = \frac{{18C}}{V} + \frac{{270C{V^3}}}{{{{\left( {150} \right)}^4}}}\)

For minimum total cost \(\frac{{d{C_t}}}{{{d_v}}} = 0\)

\(\Rightarrow \frac{{ - 18C}}{{{V^2}}} + \frac{{270C}}{{{{150}^4}}} \times 3{V^2} = 0\)

\(\Rightarrow {V^4} = \frac{{{{150}^4} \times 18}}{{270 \times 3}}\)

∴ V = 57.91 m/min