A water heater is operated by solar power. If the solar collector has an area of 3.00 m2 and the intensity delivered by sunlight is 550 W/m2, how long does it take to increase the temperature of 1.00 m3 of water from 21.0°C to 61.0°C?

To find the time it takes to heat the water, we can follow these steps:

### Step 1: Determine the Energy Needed to Heat the Water
The energy required to heat water is given by the formula:
\[
Q = mc\Delta T
\]
where:
- \( Q \) is the heat energy (in joules),
- \( m \) is the mass of the water (in kilograms),
- \( c \) is the specific heat capacity of water (\( 4186 \ \text{J/kg} \cdot \text{°C} \)),
- \( \Delta T \) is the change in temperature.

1. **Calculate the mass of water \( m \):**
   The density of water is approximately \( 1000 \ \text{kg/m}^3 \), so:
   \[
   m = \text{density} \times \text{volume} = 1000 \ \text{kg/m}^3 \times 1.00 \ \text{m}^3 = 1000 \ \text{kg}
   \]

2. **Calculate the temperature change \( \Delta T \):**
   \[
   \Delta T = 61.0^\circ \text{C} - 21.0^\circ \text{C} = 40.0^\circ \text{C}
   \]

3. **Calculate \( Q \):**
   \[
   Q = mc\Delta T = 1000 \ \text{kg} \times 4186 \ \text{J/kg} \cdot \text{°C} \times 40.0^\circ \text{C}
   \]
   \[
   Q = 167440000 \ \text{J}
   \]

### Step 2: Calculate the Power Supplied by the Solar Collector
The power \( P \) absorbed by the solar collector is given by:
\[
P = \text{intensity} \times \text{area}
\]
where:
- Intensity = \( 550 \ \text{W/m}^2 \),
- Area = \( 3.00 \ \text{m}^2 \).

Thus:
\[
P = 550 \ \text{W/m}^2 \times 3.00 \ \text{m}^2 = 1650 \ \text{W}
\]

### Step 3: Calculate the Time Required
The time \( t \) to heat the water is given by:
\[
t = \frac{Q}{P}
\]
Substitute \( Q = 167440000 \ \text{J} \) and \( P = 1650 \ \text{W} \):
\[
t = \frac{167440000 \ \text{J}}{1650 \ \text{W}}
\]
\[
t \approx 101478 \ \text{seconds}
\]

### Step 4: Convert Time to Hours
To convert seconds to hours:
\[
t = \frac{101478 \ \text{s}}{3600 \ \text{s/h}} \approx 28.2 \ \text{hours}
\]

### Final Answer
The time required to heat the water is approximately **28.2 hours**.

Would you like a breakdown of each step or have any further questions?

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Here are some related questions for practice:

1. How would the required time change if the solar collector area were doubled?
2. What would happen if the water volume were 2 m³ instead of 1 m³?
3. How would the time be affected if the sunlight intensity increased to 700 W/m²?
4. If the water started at 15°C instead of 21°C, how long would it take to reach 61°C?
5. What would the energy requirement be if the target temperature was 80°C instead of 61°C?

**Tip:** Always check unit conversions carefully, especially when working with energy and time in physics problems.

A water heater is operated by solar power. If the solar collector has an area of 3.00 m² and the intensity delivered by sunlight is 550 W/m², how long does it take to increase the temperature of 1.00 m³ of water from 21.0°C to 61.0°C?

Discover how long it takes to heat 1 m³ of water from 21.0°C to 61.0°C using a solar-powered water heater with a 3.00 m² collector and 550 W/m² sunlight intensity. This problem explores energy, power, and thermodynamic calculations suitable for Grades 11-12.

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