Question: Angela is an electrical engineer who is testing the voltage of a circuit given a certain current and resistance. She uses the following formula to calculate voltage: voltage = (current).(resistance) The circuit she tests has a current of $3 + j4$ amps and a resistance of $4 - j6$ ohms. What is the voltage of the circuit? 36 - j2 volts 36 + j34 volts - 12 - j2 volts - 12 + j34 volts
1. **State the problem:** Calculate the voltage of a circuit given the current $3 + j4$ amps and resistance $4 - j6$ ohms using the formula $$\text{voltage} = \text{current} \times \text{resistance}.$$
2. **Write the formula:** $$V = I \times R$$ where $I = 3 + j4$ and $R = 4 - j6$.
3. **Multiply the complex numbers:**
$$V = (3 + j4)(4 - j6)$$
Use distributive property:
$$= 3 \times 4 + 3 \times (-j6) + j4 \times 4 + j4 \times (-j6)$$
$$= 12 - j18 + j16 - j^2 24$$
4. **Simplify using $j^2 = -1$:**
$$= 12 - j18 + j16 - (-1)24$$
$$= 12 - j18 + j16 + 24$$
5. **Combine like terms:**
Real part: $12 + 24 = 36$
Imaginary part: $-18 + 16 = -2$
6. **Final voltage:**
$$V = 36 - j2$$ volts.
7. **Answer:** The voltage of the circuit is $36 - j2$ volts.