Question

What would be the maximum cost of a CFL such that the total cost (investment plus operating) would be the same for both CFL and incandescent $60W$ bulbs? Assume the cost of the incandescent bulb is $25cents$ and that electricity costs $10cents/kW/h$. Calculate the cost for $1000hours$, as in the cost effectiveness of CFL example.

Short answer

The price of CFL is$\mathrm{\$}4.75$, the cost of energy for CFL is $\mathrm{\$}1.5$, and the cost of energy for the incandescent bulb is $\mathrm{\$}6.0$.

Step-by-step solution

Definition of Power

The power of a process is the amount of some type of energy converted into a different type divided by the time interval $\mathrm{\Delta }t$ in which the process occurred:

$P=\frac{\mathrm{\Delta }E}{\mathrm{\Delta }t}......(1)$

The SI unit of power is the watt$(W)$.

$1$ Watt is $1$ joule/second$(1W=1J/s)$.

The given data

• The power consumed by the incandescent bulbs is:$P_{incan}=60W$.
• The power consumed by a CFL is: $P_{CFL}=15W$.
• The price of the incandescent bulb is: Price of incandescent bulb$=25cents$.
• The cost of electricity is: Electricity cost $10cents/kW/h$.

Calculation of energy consumed

The amount of energy consumed by the incandescent bulb in $1000$ hours is found from Equation $(1)$:

$E_{incan}=P_{incan}\mathrm{\Delta }t$

Substituting the values of $P_{incan}$ and $\mathrm{\Delta }t$:

$\begin{array}{rl}{E}_{incan}& =(60W)(1000h)\\ & =60\times {10}^{3}W/h\\ & =60kW/h\end{array}$

The energy consumed $60kW/h$.

Calculation of cost

The cost of this amount of energy is found by multiplying $E_{incan}$by the cost of each $1kW/h$.

$\begin{array}{rl}\mathrm{Cos}tofenergyforincandescentbulb& =(60kW/h)(10cents/kW/h\\ & =600cents\\ & =(600cents)\left(\frac{\mathrm{\$}1}{100cents}\right)\\ & =\mathrm{\$}6.0\end{array}$

Therefore, the cost of energy for the incandescent bulb is $\mathrm{\$}6.0$.

Calculation of energy consumed by CFL

The amount of energy consumed by the CFL in $1000$ hours is found from Equation $(1)$:

$E_{CFL}=P_{CFL}\mathrm{\Delta }t$

Substituting the values for $P_{CFL}$and $\mathrm{\Delta }t$:

$\begin{array}{rl}{E}_{CFL}& =(15W)(1000h)\\ & =15\times {10}^{3}W/h\\ & =15kW/h\end{array}$

The energy consumed by CFL $15kW/h$.

Calculation of energy cost for CFL

The cost of this amount of energy is found by multiplying $E_{CFL}$by the cost of each $1kW/h$:

$\begin{array}{rl}\mathrm{Cos}tofenergyforCFL& =(15kW/h)(10cents/kW/h)\\ & =150cents\\ & =(150cents)\left(\frac{\mathrm{\$}1}{100cents}\right)\\ & =\mathrm{\$}1.5\end{array}$

Therefore, the cost for CFL is $\mathrm{\$}1.5$..

Calculation of price of CFL

For the total cost of the CFL and the incandescent bulb to be the same, the price plus the electricity cost of both bulbs must be the same.

$\begin{array}{r}\mathrm{Cos}tofenergyforincandescentbulb+Priceofincandescentbulb\\ =\mathrm{Cos}tofenergyforCFL+PriceofCFL\\ \mathrm{\$}6.0+\mathrm{\$}0.25=\mathrm{\$}1.5+PriceofCFL\\ PriceofCFL=\mathrm{\$}4.75\end{array}$

Therefore, the price of CFL is $\mathrm{\$}4.75$, the cost of energy for CFL is $\mathrm{\$}1.5$ , and the cost of energy for the incandescent bulb is $\mathrm{\$}6.0$.