Question

A wire of length $20 m$ is to be cut into two pieces A piece of length $l_1$ is bent to make a square of area $A_1$ and the other piece of length $l_2$ is made into a circle of area $A_2$ If $2 A_1+3 A_2$ is minimum then $\left(\pi l_1\right): l_2$ is equal to:

• A. $6: 1$
• B. $3: 1$
• C. $4: 1$
• D. $1: 6$

Answer 1

Answer: (A)

$6: 1$

Answer 2

ℓ1​+ℓ2​=20⇒dℓ1​dℓ2​​=−1
A1​=(4ℓ1​​)2 and A2​=π(2πℓ2​​)2
Let S=2A1​+3A2​=8ℓ12​​+4π3ℓ22​​
dℓds​=0⇒82ℓ1​​+4π6ℓ2​​⋅dℓ1​dℓ2​​=0
⇒4ℓ1​​=4π6ℓ2​​⇒ℓ2​πℓ1​​=6