The equation for power in an electrical circuit is given as:

$$P = \frac{V^2}{R}$$

An electrical engineer measures the power in two circuits:

• The first circuit has a voltage of $v$ Volts and a resistance of $r$ Ohms.
• The second circuit has a voltage of $12v$ Volts and a resistance of $3r$ Ohms.

We need to find the ratio of the power in the second circuit to the power in the first circuit.

Solution:

The power in the first circuit is: $$P_1 = \frac{v^2}{r}$$

The power in the second circuit is: $$P_2 = \frac{(12v)^2}{3r}$$ Simplify $P_2$: $$P_2 = \frac{144v^2}{3r}$$ $$P_2 = 48 \cdot \frac{v^2}{r}$$

The ratio of the power in the second circuit to the power in the first circuit is: $$\frac{P_2}{P_1} = \frac{48 \cdot \frac{v^2}{r}}{\frac{v^2}{r}} = 48$$

Thus, the ratio is $48:1$.

Answer: C) $48:1$