Question

20 is divided into two parts so that the product of the cube of one part and the square of the other part is maximum, then these two parts are

• A. 15, 5
• B. 16, 4
• C. 12, 8
• D. 14, 6

Hint:

For $x+y=S$, $x^m y^n$ is max when the parts are in the ratio $m:n$. Here $3:2 \implies \frac{3}{5}(20)=12$ and $\frac{2}{5}(20)=8$.

Answer 1

The Correct Option is C

Step 1: Concept
This is an optimization problem. Let the parts be $x$ and $y$, where $x + y = 20$. We want to maximize $P = x^3 y^2$.

Step 2: Meaning
Substitute $y = 20 - x$: $P(x) = x^3(20-x)^2$.

Step 3: Analysis
Differentiate: $P'(x) = 3x^2(20-x)^2 + x^3 \cdot 2(20-x)(-1)$. Set $P'(x) = 0 \implies x^2(20-x)[3(20-x) - 2x] = 0$. $60 - 3x - 2x = 0 \implies 5x = 60 \implies x = 12$. Then $y = 20 - 12 = 8$.

Step 4: Conclusion
The two parts are 12 and 8. Final Answer: (C)