GATE 2021 CH – Question 64
Operating labor requirements L in the chemical process industry is described in terms of the plant capacity C (kg/day) over a wide range $(10^3–10^6)$ by a power law relationship $L=\alpha C^\beta$, where $\alpha$ and $\beta$ are constants. It is known that L is 60 when C is $2\times10^4$, and L is 70 when C is $6\times10^4$. The value of L when C is $10^5$ kg/day is _____ (round off to nearest integer).
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Correct answer: 75
Explanation
Divide the two known power-law equations to eliminate $\alpha$:
$$\frac{70}{60}=\left(\frac{60000}{20000}\right)^\beta=3^\beta,\qquad \beta=\frac{\ln(7/6)}{\ln3}=0.140314.$$
Scale from the first operating point:
$$L=60\left(\frac{100000}{20000}\right)^\beta=60(5)^\beta=75.2015.$$
Rounded to the nearest integer, **L = 75**. A linear interpolation in capacity would not respect the stated power law.