To solve this problem, we need to use the relationship between the instantaneous power and the phase angle in an AC circuit.

The formula for power at a phase angle is: $P = P_{\text{max}} \cos^2(\theta)$ where:

• $P$ is the power at a specific phase angle.
• $P_{\text{max}}$ is the maximum power.
• $\theta$ is the phase angle.

Here, $P = 30$ W and $\theta = 30^\circ$.

Step 1: Rewrite the equation for $P_{\text{max}}$:

$P_{\text{max}} = \frac{P}{\cos^2(\theta)}$

Step 2: Compute $\cos(30^\circ)$:

$\cos(30^\circ) = \frac{\sqrt{3}}{2}$ $\cos^2(30^\circ) = \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4}$

Step 3: Substitute values into the equation:

$P_{\text{max}} = \frac{30}{\frac{3}{4}} = 30 \times \frac{4}{3} = 40 \, \text{W}.$

Final Answer:

The maximum power ($P_{\text{max}}$) is 40 watts.

Would you like more details about this calculation or related concepts?

Related Questions:

1. What is the general formula for power in an AC circuit with resistive and reactive components?
2. How does the phase angle affect power in a circuit with inductors and capacitors?
3. What is the significance of the maximum power in AC circuits?
4. How can we derive $\cos(30^\circ)$ geometrically or algebraically?
5. What happens to power if the phase angle changes to $60^\circ$?

Tip:

When working with AC power calculations, always double-check trigonometric identities and their squared values for precise results.