City Guilds Model Questions Vol1 2019 Engineering Question 10

A turning operation follows the tool‑life equation \(T = \dfrac{C}{V^{n} f^{m}}\) where \(T\) is tool life in minutes, \(V\) is cutting speed (m·min\(^{-1}\)), \(f\) is feed (mm·rev\(^{-1}\)), \(C = 1.2\times10^{5}\), \(n = 0.25\) and \(m = 0.5\). The workpiece requires removal of \(0.12\) m\(^3\). The workpiece diameter is \(D = 80\) mm. Material removal rate is \(\text{MRR}=\pi D f N\) where \(N\) is spindle speed (rev·min\(^{-1}\)). Cutting speed relates to spindle speed by \(V = \dfrac{\pi D N}{1000}\). The machine can operate at most \(V_{max}=120\) m·min\(^{-1}\) and \(f_{max}=0.30\) mm·rev\(^{-1}\). Machining cost is \(\text{₦}0.5\) per minute and each tool costs \(\text{₦}150\). Total cost = machining time × 0.5 + (number of tools used) × 150. Assuming a tool is replaced only when its life expires, determine the values of \(V\) and \(f\) (to the nearest integer for \(V\) and two decimal places for \(f\)) that minimise the total cost, and give that minimum cost (rounded to the nearest naira).