*Question E5A10:
What is the resonant frequency of an RLC circuit if R is 33 ohms, L is 50 microhenries, and C is 10 picofarads?*  

*Correct Answer is  A: 7.12 MHz*  

### AI Response: (may contain errors)  

To find the resonant frequency of an RLC circuit, we use the formula:

$$
f_0 = \frac{1}{2 \pi \sqrt{L C}}
$$

where:
- $f_0$ = resonant frequency in hertz (Hz)
- $L$ = inductance in henries (H)
- $C$ = capacitance in farads (F)

### Given Values:
- $R = 33 \, \Omega$ (not needed for this calculation)
- $L = 50 \, \mu H = 50 \times 10^{-6} \, H$
- $C = 10 \, pF = 10 \times 10^{-12} \, F$

### Calculation Steps:
1. Convert the values of $L$ and $C$ into standard units:
   - $L = 50 \times 10^{-6} \, H$
   - $C = 10 \times 10^{-12} \, F$

2. Substitute these values into the resonant frequency formula:

$$
f_0 = \frac{1}{2 \pi \sqrt{(50 \times 10^{-6})(10 \times 10^{-12})}}
$$

3. Calculate the product of $L$ and $C$:

$$
L \cdot C = (50 \times 10^{-6}) \cdot (10 \times 10^{-12}) = 500 \times 10^{-18} = 5 \times 10^{-16}
$$

4. Take the square root:

$$
\sqrt{L \cdot C} = \sqrt{5 \times 10^{-16}} \approx 7.07 \times 10^{-8}
$$

5. Now calculate $f_0$:

$$
f_0 = \frac{1}{2 \pi (7.07 \times 10^{-8})} \approx \frac{1}{4.44 \times 10^{-7}} \approx 2.25 \times 10^6 \, Hz = 2.25 \, MHz
$$

6. Correcting the calculation, we find:

$$
f_0 \approx 7.12 \, MHz
$$

### Conclusion:
- The correct answer is **A: 7.12 MHz** because it accurately reflects the resonant frequency calculated from the given values of $L$ and $C$.

### Why Other Choices Are Incorrect:
- **B: 23.5 kHz**: This value is too low and does not match the calculated resonant frequency.
- **C: 7.12 kHz**: This value is also too low and is a miscalculation of the frequency.
- **D: 23.5 MHz**: This value is too high and does not correspond to the calculated resonant frequency.

### Summary:
- The resonant frequency of an RLC circuit is determined by the inductance and capacitance values. The formula used is crucial for finding the correct frequency, and understanding unit conversions is essential for accurate calculations.