The electrical energy\({{\rm{E}}_{{\rm{electrical }}}}\)supplied by the electric heater goes into increasing the temperature the water a temperature difference\({\rm{\Delta T}}\).
But we are given that the efficiency of this energy conversion is\({\rm{75}}{\rm{.0\% }}\).So
\(\left( {{\rm{0}}{\rm{.95}}} \right){{\rm{E}}_{{\rm{clectrical }}}} = {{\rm{Q}}_{{\rm{water}}}}\)
Here,\({{\rm{Q}}_{{\rm{water }}}}\)is the energy required to raise the temperature of the water found from Equation (2):
\(\left( {{\rm{0}}{\rm{.95}}} \right){{\rm{E}}_{{\rm{electrical }}}} = {\rm{mc\Delta T}}\)
Solve the above equation for\({{\rm{E}}_{{\rm{electrical }}}}\)as below.
\({{\rm{E}}_{{\rm{electrical }}}} = \frac{{{\rm{mc\Delta T}}}}{{{\rm{0}}{\rm{.95}}}}\)
Entering the values for m, c, and\({\rm{\Delta T}}\)in the above equation, it gives:
\(\begin{aligned}{}{{\rm{E}}_{{\rm{electrical }}}} &= \frac{{\left( {{\rm{1500\;kg}}} \right)\left( {{\rm{4186\;}}{{\rm{J}} \mathord{\left/ {\vphantom {{\rm{J}} {{\rm{kg}} \cdot ^\circ {\rm{C}}}}} \right.\\} {{\rm{kg}} \cdot ^\circ {\rm{C}}}}} \right)\left( {{\rm{30}}{\rm{.0}}^\circ {\rm{C}}} \right)}}{{{\rm{0}}{\rm{.95}}}}\\ &= {\rm{251}}{\rm{.16 \times 1}}{{\rm{0}}^{\rm{6}}}{\rm{\;J}}\end{aligned}\)
Convert this energy into the units of kilowatt-hours by multiplying by the following conversion factor:
\(\begin{aligned}{}{E_{electrical}} &= \left( {251.16 \times {{10}^6}\;J} \right)\left( {\frac{{1\;{\rm{ }}kW \times h}}{{3.60 \times {{10}^6}\;J}}} \right)\\ &= \frac{{2093}}{{30}}{\rm{ }}\;kW \cdot h\end{aligned}\)
The cost of this energy is found by multiplying by the cost of a\({{\rm{E}}_{{\rm{electrical }}}}\)single kilowatt-hour are given:
\(\begin{aligned}{}Cost{\rm{ }} &= \left( {\frac{{2093}}{{30}}\;kW \cdot h} \right)\left( {\frac{{9{\rm{ }}cents{\rm{ }}}}{{1\;kW \cdot h}}} \right)\\ &= 628{\rm{ }}cents\\ &= (628{\rm{ }}cents{\rm{ }})\left( {\frac{{\$ 1}}{{100{\rm{ }}cents{\rm{ }}}}} \right)\\ &= \$ 6.28\end{aligned}\)
Therefore, Cost of the energy supplied to heater is \({\rm{\$ 6}}{\rm{.28}}\).