Calculate the power wasted through a cable of resistance 0.5 \ when 11 \ kW electricity generated is transmitted through it at 120 \ V.

Here are the solutions to your problems.

** (Power wasted in a cable):**
Calculate the power wasted through a cable of resistance $0.5 \, \Omega$ when $11 \, \text{kW}$ electricity generated is transmitted through it at $120 \, \text{V}$.

Step 1: Identify the given values.
Resistance of the cable ($R$) = $0.5 \, \Omega$.
Electricity generated ($P_{gen}$) = $11 \, \text{kW} = 11000 \, \text{W}$.
Transmission voltage ($V$) = $120 \, \text{V}$.

Step 2: Calculate the current flowing through the cable during transmission.
The current ($I$) can be found using the formula $P = V I$.

$$I = \frac{P_{gen}}{V}$$ $$I = \frac{11000 \, \text{W}}{120 \, \text{V}}$$ $$I \approx 91.6667 \, \text{A}$$

Step 3: Calculate the power wasted in the cable.
The power wasted ($P_{wasted}$) due to the resistance of the cable is given by $P_{wasted} = I^2 R$.

$$P_{wasted} = (91.6667 \, \text{A})^2 \times 0.5 \, \Omega$$ $$P_{wasted} = 8402.78 \times 0.5 \, \text{W}$$ $$P_{wasted} = 4201.39 \, \text{W}$$

To express in kilowatts:

$$P_{wasted} = \frac{4201.39}{1000} \, \text{kW}$$ $$P_{wasted} \approx 4.20 \, \text{kW}$$

The power wasted in the cable is $\boxed{\text{4.20 kW}}$.
Note: The provided answer in the image (1.05 KW) does not match the calculation based on the given values.

** (Resonance frequency):**
A $10 \, \text{mF}$ capacitor is connected in series with a coil of $2 \, \text{mH}$ inductance and $5.0 \, \Omega$ resistance. Determine the resonance frequency.

Step 1: Identify the given values.
Capacitance ($C$) = $10 \, \text{mF} = 10 \times 10^{-3} \, \text{F}$.
Inductance ($L$) = $2 \, \text{mH} = 2 \times 10^{-3} \, \text{H}$.
Resistance ($R$) = $5.0 \, \Omega$. (Note: Resistance does not affect the resonance frequency itself in a series RLC circuit).

Step 2: Use the formula for resonance frequency ($f_0$) in a series RLC circuit.

$$f_0 = \frac{1}{2\pi\sqrt{LC}}$$

Step 3: Substitute the values and calculate the resonance frequency.

$$f_0 = \frac{1}{2\pi\sqrt{(2 \times 10^{-3} \, \text{H})(10 \times 10^{-3} \, \text{F})}}$$ $$f_0 = \frac{1}{2\pi\sqrt{20 \times 10^{-6} \, \text{H} \cdot \text{F}}}$$ $$f_0 = \frac{1}{2\pi \times \sqrt{20} \times 10^{-3}}$$ $$$$