Question

A house has four tube lights, three fans and a television. Each tubelight draws 40 W. The fan draws 80 W and the television draws 100 W. On an average, all the tubelights are kept on for 5 hours, all fans for 12 hours and the television for 6 hours every day. Find the cost of the electric energy used in 30 days at the rate of Rs. 3.00 per kWh

Answer 1

To solve this problem, we have to establish the definitions and the relationship between the quantities, energy and power.
As per Joule’s law of heating, the heat produced in the circuit is given by –
$H = {I^2}Rt$
where $I$= current in the circuit, R = resistance in the circuit and t is the time
The power is defined as the heat produced in the circuit per unit time. Thus,
$P = \dfrac{H}{t} = {I^2}R$

Complete step-by-step answer:
The unit of power is watt, where one watt is defined as one joule of heat produced in one second.
1 watt = 1 joule/second.
Now, we use this unit to derive another unit of energy known as kilowatt-hour (kWh) which is the commercial unit of energy which is used in distribution.
$1kWh = 1000W \times 1hr = 1000 \times 3600 = 3 \cdot 6 \times {10^6}J$
In the house, there are 4 tube lights, 3 fans and 1 television.
Each tube light consumes 40 W and switches on for 5 hours. Hence, the total electric energy consumption by the 4 tube lights in kWh is:
${e_l} = 4 \times 40 \times 5 = 800Wh = 0 \cdot 8kWh$
This is the energy consumption per day. For 30 days, the tube lights consume,
${E_l} = {e_l} \times 30 = 0 \cdot 8 \times 30 = 24kWh$
Each fan consumes 80 W and switches on for 12 hours. Hence, the total electric energy consumption by the 3 fans in kWh is:
${e_f} = 3 \times 80 \times 12 = 2880Wh = 2 \cdot 88kWh$
This is the energy consumption per day. For 30 days, the fans consume,
${E_f} = {e_f} \times 30 = 2 \cdot 88 \times 30 = 86 \cdot 4kWh$

The television consumes 100 W and switches on for 6 hours. Hence, the total electric energy consumption by the television in kWh is:
${e_t} = 100 \times 6 = 600Wh = 0 \cdot 6kWh$
This is the energy consumption per day. For 30 days, the television consumes,
${E_t} = {e_t} \times 30 = 0 \cdot 6 \times 30 = 18kWh$
The net energy consumption, $E = {E_l} + {E_f} + {E_t}$
Substituting, we get –
$E = 24 + 86 \cdot 4 + 18 = 128 \cdot 4kWh$
The rate of electricity is Rs. 3.00. The total cost of the electric energy,
$C = E \times 3 = 128 \cdot 4 \times 3 = 385 \cdot 2$

Hence, the cost of electric energy is Rs. 385.2

Note: The students might have a general doubt as to why the commercial units of energy used for calculating the cost, is in kilo-watt hour and not in joules, since the joule is the standard unit of energy.
For clearing that doubt, let us convert the obtained answer for energy here, into joules.
$E = 128 \cdot 4kWh$
The conversion is given by:
$1kWh = 3 \cdot 6 \times {10^6}J$
Converting, we get –
$E = 128 \cdot 4kWh = 128 \cdot 4 \times 3 \cdot 6 \times {10^6} = 0 \cdot 4 \times {10^9}J$
Here you can see that we get an exceedingly high value in joules. This cannot be very convenient for using to calculate the costs. In order to simplify the process, the unit kilo-watt hour is in use for commercial purposes.