The firm uses the following function to calculate the production output (PO): PO $= 5.3 C^2 L^{15}$, where $C$ = capital invested and $L$ = labour employed. If the capital invested ($C$) increases by $20$ percent, the change in PO will be

### Concept & Formula When analyzing proportionate changes in an equation, constants (like $5.3$) and unchanged variables (like $L$) can be ignored. The output $PO$ is directly proportional to the square of the capital $C$. $$PO \propto C^2$$ If $C$ scales by a factor, $PO$ scales by the square of that factor. ### Step-by-step Solution **Step 1: Identify the multiplying factor for C** Capital $C$ increases by $20\%$. New Capital $C_{new} = C + 0.20C = 1.20C$ **Step 2: Apply the change to the proportionality** Since $PO \propto C^2$, we square the new factor to find the effect on $PO$. $PO_{new} \propto (1.20C)^2$ $PO_{new} \propto 1.44 C^2$ **Step 3: Calculate the percentage change** The original factor was $1.0$. The new factor is $1.44$. Change = $1.44 - 1.0 = 0.44$ To convert to a percentage, multiply by $100$: $0.44 \times 100 = 44\%$ increase. ### Exam Strategy & Shortcut Use the successive percentage increase formula for squared variables: $$\text{Net Change} = a + b + \frac{ab}{100}$$ Since $C$ is squared, it acts as two identical successive increases of $20\%$. Net Change $= 20 + 20 + \frac{20 \times 20}{100}$ Net Change $= 40 + 4 = 44\%$ increase. This skips algebraic substitution entirely and solves the problem in seconds. ### Common Pitfall A very common mistake is assuming a $20\%$ increase in a variable leads to exactly a $20\%$ increase in the output, ignoring the exponent. Always check the power of the changing variable in the equation. ### Final Answer **Therefore, the correct answer is 44% increase.**