How to Configure a Circuit for a 1600 W Radio Device with Given Resistors?

[τheory]

Homework Statement

You are given a battery of 220 V and a radio device that requires 1600 W to run. You are also given four resistors: three 3 Ω resistors and one 20 Ω resistor. What circuit configuration will allow the radio device to run without getting fried?

The Attempt at a Solution

Power = (Voltage)²/Resistance
1600 W = (220 V)²/R
R = 30.25 Ω

So this device would require 30.25 Ω to function correctly. However, based on several circuit configurations, this magnitude of resistance isn't obtainable? For example, if I used just a complete series circuit, then the total resistance would be:

R total = R1 + R2 + R3 + R4
R total = 3 + 3 + 3 + 20 = 29 Ω total which isn't exactly 30.25

Incorporating a parallel circuit would cause resistance values to be even less than their original values because parallel circuits require the reciprocal resistance values. So if I had a circuit consisting of all the resistors parallel, then I would get:

1/R total = 1/R1 + 1/R2 + 1/R3 + 1/R4
1/R total = 1/3 + 1/3 + 1/3 + 1/20
1/R total = 0.95 Ω, which is way less

Thus, can someone point me in the right direction?

 

[cupid.callin]

Ummm ... The question doesn't say that radio works at 220V.
Right? :p

 

[τheory]

What do you mean?

 

[Zubeen]

You made certain mistakes regarding the use of the formula
like-

Power = (Voltage)²/Resistance
1600 W = (220 V)²/R
R = 30.25 Ω

Since the Resistance is to be attached to the circuit , the V=220 V will be wrong as there will be potential drop across the resistor (if attached in series to the radio).
So we must go like this,
Let V₁ be the P.D (potential drop) across the resistor combination (R₀ ), and V₂ be potential drop across the radio.
Then we have that V₁ + V₂ = 220.
so since its a series combination of net resistance of the combination and the radio, the current through both of them will be the same.
then,
we have
IR₀ + 1600/I =220
on solving , you will get a quadratic equation in terms of I.
Further, since I can be real number only so the D of quadratic equation must be >0
that will give =>
R₀ < 121/16
so any combination satisfying above inequality will be the result and similarly , you can check for the parallel combination of radio and net resistance.


Zubeen