Question

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