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Let's break down the problem and solve it step by step.

### Given Information:
- The light bulb has a resistance of \( R = 144 \, \Omega \).
- The voltage varies according to \( V(t) = V_0 \sin(2 \pi f t) \), where:
  - \( V_0 = 110 \, \text{V} \) (maximum voltage),
  - \( f = 60 \, \text{Hz} \) (frequency of the voltage signal).
- The time period \( T = 5 \, \text{hours} = 5 \times 3600 = 18,000 \, \text{seconds} \).
- We are asked to calculate the total energy \( U \) in joules and interpret the result.

### Formula:
The energy \( U \) expended by the bulb over time is given by:
\[
U = \int_0^T P(t) \, dt
\]
where \( P(t) = \frac{V(t)^2}{R} \) is the power at time \( t \), and \( V(t) = V_0 \sin(2 \pi f t) \).

### Step-by-Step Solution:

1. **Expression for \( P(t) \):**
   \[
   P(t) = \frac{V(t)^2}{R} = \frac{V_0^2 \sin^2(2 \pi f t)}{R}
   \]
   Substituting the given values:
   \[
   P(t) = \frac{110^2 \sin^2(2 \pi \cdot 60 \cdot t)}{144}
   \]
   Simplifying:
   \[
   P(t) = \frac{12100 \sin^2(120 \pi t)}{144}
   \]
   \[
   P(t) = 84.03 \sin^2(120 \pi t) \, \text{watts}
   \]

2. **Average Power over Time:**
   Since \( \sin^2(x) \) has a time-average of \( \frac{1}{2} \), we can calculate the average power:
   \[
   P_{\text{avg}} = 84.03 \times \frac{1}{2} = 42.015 \, \text{watts}
   \]

3. **Energy Calculation:**
   Energy is the power multiplied by time:
   \[
   U = P_{\text{avg}} \times T
   \]
   Substituting \( T = 18,000 \, \text{seconds} \):
   \[
   U = 42.015 \times 18,000 = 756,270 \, \text{joules}
   \]

### Interpretation of Result:
The total energy expended by the 100W light bulb over 5 hours is approximately **756,270 joules**. This means the bulb consumes around 756.27 kJ of energy during this period, consistent with typical household light bulb usage.

Would you like more details or have any questions? 

Here are 5 related questions to think about:
1. How would the energy change if the resistance of the bulb were higher or lower?
2. What happens to the total energy consumption if the voltage fluctuates more or less?
3. How does the frequency of the voltage source affect the energy consumed?
4. Can we calculate the energy in kilowatt-hours (kWh) and how would that compare to joules?
5. How would you calculate the energy consumption if the light bulb were turned on and off intermittently?

**Tip:** The relationship between voltage, resistance, and power is crucial for analyzing electrical systems, especially when voltage varies over time like in alternating current (AC) systems.

A 100W light bulb has resistance R = 144Ω (ohms) when attached to household current, where the voltage varies as V = V0sin(2πft), where V0 = 110V and f = 60Hz. The energy (in joules) expended by the bulb over a period of T seconds is U = ∫₀ᵀ P(t) dt, where P = V²/R (J/s) is the power. Compute U if the bulb remains on for 5 hours (include the appropriate units), and write a statement interpreting your result.

Solve the problem of computing the total energy expended by a 100W light bulb with resistance of 144Ω when subjected to a varying voltage over 5 hours. This detailed solution covers key concepts such as sinusoidal voltage, power, and time-averaged power. Includes a step-by-step solution with units and interpretation.

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