Two variable resistors R₁ = 2x and R₂ = 32 x (where x > 0 ) are connected in parallel. An ideal 24 V DC voltage source is connected across this combination. The minimum total power dissipated in the circuit is _____ W .

Step-by-step solution

The equivalent resistance R_ eq of the parallel combination is given by: R_ eq = R₁ R₂ R₁ + R₂ = (2x) ( 32 x ) 2x + 32 x = 64 2x + 32 x The total power dissipated in the circuit is: P = V^2 R_ eq = (24)^2 64 2x + 32 x = 576 64 (2x + 32 x ) = 9 (2x + 32 x ) To minimize the power P , the term (2x + 32 x ) must be minimized. Using the AM-GM inequality for the positive terms 2x and 32 x : 2x + 32 x 2 2x 32 x 2x + 32 x 2 64 = 16 The minimum value of the expression is 16 . Therefore, the minimum power dissipated is: P_ m