To solve the problem of determining the time it takes to heat water using two heater coils, we need to consider configurations of the coils connected in series and in parallel. Let's go through the steps for both cases.
### Heaters Connected in Series
When the heaters are connected in series, the total heating time is the sum of the times taken by each individual heater. If one heater takes time [tex]\( t_1 \)[/tex] and the other takes time [tex]\( t_2 \)[/tex] to heat a glass of water, the total time [tex]\( T_{\text{series}} \)[/tex] when connected in series is simply:
[tex]\[ T_{\text{series}} = t_1 + t_2 \][/tex]
### Heaters Connected in Parallel
When the heaters are connected in parallel, they provide their heating power simultaneously. To find the effective heating time [tex]\( T_{\text{parallel}} \)[/tex] when the heaters are connected in parallel, we need to consider their combined rate of heating. The rate of heating by each heater can be represented as the reciprocal of the time they take:
- Rate of heater 1: [tex]\( \frac{1}{t_1} \)[/tex]
- Rate of heater 2: [tex]\( \frac{1}{t_2} \)[/tex]
The combined rate of heating when both heaters are connected in parallel is the sum of these individual rates:
[tex]\[ \text{Combined rate} = \left( \frac{1}{t_1} + \frac{1}{t_2} \right) \][/tex]
The effective time required to heat the glass of water when both heaters are connected in parallel is the reciprocal of this combined rate:
[tex]\[ T_{\text{parallel}} = \frac{1}{\left( \frac{1}{t_1} + \frac{1}{t_2} \right)} \][/tex]
### Example Calculation
Given specific times:
- [tex]\( t_1 = 10 \)[/tex] minutes for the first heater
- [tex]\( t_2 = 20 \)[/tex] minutes for the second heater
The time when connected in series:
[tex]\[ T_{\text{series}} = t_1 + t_2 = 10 + 20 = 30 \text{ minutes} \][/tex]
For the parallel connection:
1. Calculate the individual rates:
[tex]\[ \frac{1}{t_1} = \frac{1}{10} \][/tex]
[tex]\[ \frac{1}{t_2} = \frac{1}{20} \][/tex]
2. Sum the rates:
[tex]\[ \text{Combined rate} = \frac{1}{10} + \frac{1}{20} = \frac{2}{20} + \frac{1}{20} = \frac{3}{20} \][/tex]
3. Take the reciprocal of the combined rate to find the time:
[tex]\[ T_{\text{parallel}} = \frac{1}{\left( \frac{3}{20} \right)} = \frac{20}{3} \approx 6.6667 \text{ minutes} \][/tex]
### Final Answer
- When connected in series, the time taken will be [tex]\( 30 \)[/tex] minutes.
- When connected in parallel, the time taken will be approximately [tex]\( 6.67 \)[/tex] minutes.
