A circuit consists of two sections connected in series. Section A has two identical resistors, each of resistance xR₀ , connected in parallel. Section B consists of a single resistor of resistance 8R₀ x . Here x is a variable positive parameter and R₀ is a constant resistance. If the minimum possible equivalent resistance of the entire circuit is 48 , the value of R₀ is _____ .

Step-by-step solution

The equivalent resistance of Section A, which has two identical resistors in parallel, is: R_A = xR₀ 2 The resistance of Section B is: R_B = 8R₀ x Since Section A and Section B are connected in series, the total equivalent resistance R_ eq of the circuit is: R_ eq = R_A + R_B = xR₀ 2 + 8R₀ x To find the minimum value of R_ eq , we can use the AM-GM inequality for the two positive terms: xR₀ 2 + 8R₀ x 2 xR₀ 2 8R₀ x R_ eq 2 4R₀^2 = 4R₀ Thus, the minimum possible equivalent resistance is 4R₀ . We are given that this m