Question

The resistance of a device component decreases as the current through it increases and it is described by the relation, $R=\frac{0.2 \,I}{I-4}$, where $I$ is the current. Determined the minimum power deliver. (Assume, $I>4$ )

• A. 22.4 W
• B. 18.6 W
• C. 19.8 W
• D. 21.6 W

Answer 1

Answer: (D)

Given, resistance $R$ of a device decreases as current, $I$ increases by the relation,
$R=\frac{0.2\, I}{I-4}$
$\therefore $ Power of the electric device, $P=I^{2} R$
$=I^{2} \cdot \frac{0.2\, I}{I-4}$
Power will be minimum, if
$\frac{d P}{d I}=0$
$\frac{d}{d I}\left(\frac{0.2\, I^{3}}{I-4}\right)=0$
$\frac{(I-4) \times 0.2 \times 3\, I^{2}-0.2 \, I^{3}}{(I-4)^{2}}=0$
$0.6 \, I^{3}-2.4\, I^{2}-0.2\, I^{3}=0$
$0.4\, I^{3}-2.4\, I^{2}=0$
$0.4 \, I^{2}(I-6)=0\,\,(\because I>4)$
$ \Rightarrow I-6=0$
$I=6\, A$
From Eq. (i), minimum power
$P_{\min }=\frac{0.2 \times 6^{3}}{6-4}=\frac{43.2}{2}=21.6 \, W$
Hence, the minimum power delivered in a device $21.6 \, W$.