Applications of Derivatives
Optimization — Wire Cut into Square and Circle
nta_pyq_2023_jan
Grade None
Question:
A wire of length 20 m is to be cut into two pieces. A piece of length $\ell_1$ is bent to make a square of area $A_1$ and the other piece of length $\ell_2$ is made into a circle of area $A_2$. If $2A_1+3A_2$ is minimum, then $(\pi\ell_1):\ell_2$ is equal to:
Step-by-Step Solution
Key Concept: $A_1=\left(\frac{\ell_1}{4}\right)^2$, $A_2=\pi\left(\frac{\ell_2}{2\pi}\right)^2=\frac{\ell_2^2}{4\pi}$. Let $S=2A_1+3A_2=\frac{\ell_1^2}{8}+\frac{3\ell_2^2}{4\pi}$. With $\ell_1+\ell_2=20$.
$(\pi\ell_1):\ell_2=6:1$.
Correct Answer: 1