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.