To solve the problem step by step, we will follow the process of calculating the effective power of the heater, the heat required to raise the temperature of the water, and finally, the time taken to reach the desired temperature.
### Step 1: Calculate the effective power of the heater
The heater has a power of 1 kW, which is equivalent to 1000 Joules per second. However, it loses heat to the surroundings at a rate of 160 Joules per second. Therefore, we need to calculate the effective power that contributes to heating the water.
\[
\text{Effective Power} = \text{Power of Heater} - \text{Heat Lost}
\]
\[
\text{Effective Power} = 1000 \, \text{J/s} - 160 \, \text{J/s} = 840 \, \text{J/s}
\]
### Step 2: Calculate the heat required to raise the temperature of the water
We need to calculate the heat required to raise the temperature of 2 liters of water from 27°C to 77°C.
First, convert the volume of water to mass:
\[
\text{Mass of water} = 2 \, \text{liters} = 2000 \, \text{grams}
\]
Next, calculate the change in temperature:
\[
\Delta T = 77°C - 27°C = 50°C
\]
Now, using the specific heat capacity of water, which is approximately \(4.2 \, \text{J/g°C}\), we can calculate the total heat required (Q):
\[
Q = \text{mass} \times \text{specific heat} \times \Delta T
\]
\[
Q = 2000 \, \text{g} \times 4.2 \, \text{J/g°C} \times 50°C
\]
\[
Q = 2000 \times 4.2 \times 50 = 420000 \, \text{J}
\]
### Step 3: Calculate the time required to reach the desired temperature
Now, we can find the time (T) required to supply the necessary heat using the effective power calculated earlier. The formula relating power, time, and energy is:
\[
\text{Power} = \frac{Q}{T} \implies T = \frac{Q}{\text{Effective Power}}
\]
Substituting the values we have:
\[
T = \frac{420000 \, \text{J}}{840 \, \text{J/s}} = 500 \, \text{s}
\]
### Step 4: Convert time from seconds to minutes
To convert seconds into minutes:
\[
T = \frac{500 \, \text{s}}{60} \approx 8.33 \, \text{minutes}
\]
This is approximately 8 minutes and 20 seconds.
### Final Answer
The time required for the temperature to reach 77°C is approximately **8 minutes and 20 seconds**.
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