Question

An energy-efficient light bulb, taking in 28.0 W of power, can produce the same level of brightness as a conventional light bulb operating at power 100 W. The lifetime of the energy-efficient bulb is 10000 h and its purchase price is \(4.50, whereas the conventional bulb has a lifetime of 750 h and costs \)0.42. Determine the total savings obtained by using one energy-efficient bulb over its lifetime as opposed to using conventional bulbs over the same time interval. Assume an energy cost of $0.200 per kilowatt-hour.

Short answer

The total savings obtained by using one energy-efficient bulb over its lifetime as opposed to using conventional bulbs over the same time interval is:$\$145$ .

Step-by-step solution

Defining the power and energy formula

The instantaneous power P is defined as the time rate of energy transfer.

$P=\frac{dE}{dt}$

Wecan write

$Energy=Power\times time$

where

$P=$Power

$\Delta E=$Change in total energy

$\Delta t=$Change in time in seconds.

Determining the energy and power cost

Fromstep(1),wehave

$Energy=Power\times time$

For the$28.0W$ bulb

Calculating the used energy

$$\begin{gathered} E=\left(28.0w\right)\left(1.00\times {10}^{4}h\right) \\ =280kwh \end{gathered}$$

Calculating the total cost

$$\begin{gathered} C=\left(\$0.450\right)+\left(280kwh\right)\left(\$0.200/kwh\right) \\ =\$60.50 \end{gathered}$$

For the$100\text{W}$ bulb

Energy used

$$\begin{gathered} E\text{'}=\left(100w\right)\left(1.00\times {10}^{4}h\right) \\ =1.00\times {10}^{3}kwh \end{gathered}$$

Bulbs used

$$\begin{gathered} N=\frac{1.00\times {10}^{4}h}{750h/bulb} \\ =13.3 \\ \approx 13bulbs \end{gathered}$$

Total cost is

$$\begin{gathered} P=\left(13\times \$0.420\right)+\left(1.00\times {10}^{3}kwh\right)\left(\$0.200/kwh\right) \\ =\$205.46 \end{gathered}$$

Savings with energy-efficient bulb

$\$205.46-\$60.50=\$145$