The power absorbed by R_o is 250 W. Find all values of R_o for which this is true.

First, find the Thevenin equivalent of the circuit connected to R_o. I will do this by finding V_oc and I_sc.

Find V_oc. I will use mesh analysis.

Dependent source parameter:

iD = i_b

KVL around mesh a:

25 i_a + 100 (i_a - i_b) - 200 = 0

125 i_a - 100 i_b = 200 (Equation A)

KVL around mesh b:

10 i_b + 20 i_b + 30 iD + 100 (i_b - i_a) = 0

10 i_b + 20 i_b + 30 i_b + 100 (i_b - i_a) = 0

- 100 i_a + 160 i_b = 0 (Equation B)

Solving Equations A and B yields:

i_a = 3.2 A

i_b = 2 A

Thus by KVL

V_oc = 20 i_b + 30 i_b = 50 (2)

V_oc = 100 V = V_Th

Now find I_sc. Again, I will use mesh analysis.

Dependent source parameter:

iD = i_b

KVL around mesh a:

25 i_a + 100 (i_a - i_b) - 200 = 0

125 i_a - 100 i_b = 200 (Equation C)

KVL around mesh b:

10 i_b + 20 (i_b - i_c) + 30 iD + 100 (i_b - i_a) = 0

10 i_b + 20 (i_b - i_c) + 30 i_b + 100 (i_b - i_a) = 0

- 100 i_a + 160 i_b - 20 i_c = 0 (Equation D)

KVL around mesh c:

20 (i_c - i_b) - 30 iD = 0

20 (i_c - i_b) - 30 i_b = 0

- 50 i_b + 20 i_c = 0 (Equation E)

Solving Equations C, D, and E yields:

i_a = 5.867 A

i_b = 5.333 A

i_c = 13.333 A = I_sc

Thus

R_Th = V_oc / I_sc = 100 / 13.333

R_Th = 7.5 W

By voltage division,

The power absorbed by R_o is

P_o = V_o² / R_o

But we know that the power absorbed is 250 W, so

Solving for R_o yields

R_o = 2.5 W

or

R_o = 22.5 W

You might note that the value of R_o for maximum power transfer (7.5 W) lies between these two values. In a problem of this nature, this will always be the case.