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Chapter 7: Problem 41

An energy-efficient lightbulb, taking in \(28.0 \mathrm{W}\) of power, can produce the same level of brightness as a conventional bulb operating at power \(100 \mathrm{W}\). The lifetime of the energy efficient bulb is \(10000 \mathrm{h}\) and its purchase price is 17.0 dollar, whereas the conventional bulb has lifetime \(750 \mathrm{h}\) and costs 0.420 dollar per bulb. Determine the total savings obtained by using one energy-efficient bulb over its lifetime, as opposed to using conventional bulbs over the same time period. Assume an energy cost of 0.0800 dollar per kilowatt-hour.

Short Answer

Expert verified
The total saving obtained by using one energy-efficient bulb over its lifetime, as opposed to using conventional bulbs over the same time period, is $46.48.

Step by step solution

01

Calculate the life time energy cost of the energy-efficient bulb

Firstly, we need to know the total energy consumption of the energy-efficient bulb over its life time, the power of the energy-efficient bulb is \(28.0 \mathrm{W}\) which is \(0.028 \mathrm{KW}\), and the life time is \(10000 \mathrm{h}\), so the total energy used is \(0.028 \mathrm{KW} \times 10000 \mathrm{h} = 280 \mathrm{KWh}\). As energy cost is \(\$0.0800/\mathrm{KWh}\), the total energy cost over the life time is \(280 \mathrm{KWh} \times \$0.0800/\mathrm{KWh} = \$22.4\). The total cost is given by the sum of energy cost and initial purchase price, so for the energy-efficient bulb it is \(\$22.4 + \$17.0 = \$39.4\).
02

Calculate the life time cost of using conventional bulbs

As in the case of the energy-efficient bulb, we calculate the energy cost first. The power of the conventional bulb is \(100 \mathrm{W}\), or \(0.100 \mathrm{KW}\), and as each conventional bulb only lasts for \(750 \mathrm{h}\), we need \(10000 \mathrm{h} / 750 \mathrm{h} = 13.33\) bulbs to last the same period of time as an energy-efficient bulb. Here we consider that we need to buy 14 bulbs as we can't buy less than 1 bulb and we round up to ensure we have enough bulbs for the entire time period. The total energy used by this many bulbs is \(0.100 \mathrm{KW} \times 10000 \mathrm{h} = 1000 \mathrm{KWh}\), hence the total energy cost over the life time is \(1000 \mathrm{KWh} \times \$0.0800/\mathrm{KWh} = \$80.0\). Again, the total cost is given by the sum of energy cost and initial purchase price, so for the conventional bulbs it is \(\$80.0 + 14 \times \$0.420 = \$85.88\).
03

Calculate the total savings

The savings obtained by using an energy-efficient bulb is the cost of using conventional bulbs subtracted by the cost of using the energy-efficient bulb, so it is \(\$85.88 - \$39.4 = \$46.48\). So we save \$46.48 over the lifetime of an energy-efficient bulb.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Energy consumption calculation
Energy consumption is a critical factor when comparing lightbulbs. The way we determine energy usage is by considering the power rated in kilowatts (KW) multiplied by the number of hours the bulb operates. For an energy-efficient bulb that uses 28 watts, we convert to kilowatts by dividing by 1000, giving us 0.028 KW. Over its lifetime of 10,000 hours, the total energy consumption is calculated as:
\[0.028 \, \text{KW} \times 10,000 \, \text{h} = 280 \, \text{KWh}.\]
  • An energy-efficient bulb, therefore, utilizes 280 KWh over its lifetime.
  • In contrast, a conventional bulb uses 100 watts or 0.100 KW.
  • Operating this for the same 10,000 hours requires:
    \[0.100 \, \text{KW} \times 10,000 \, \text{h} = 1000 \, \text{KWh}.\]
Ultimately, energy-efficient bulbs consume less energy, reducing the environmental footprint and operational costs.
Cost savings
Understanding cost savings involves comparing the total expenses associated with each bulb type. This includes both the purchase price and the energy cost over the bulb's lifetime. An energy-efficient bulb costs \(17 to purchase. With energy priced at \)0.0800 per KWh, the bulb's total energy cost becomes:
\[280 \, \text{KWh} \times 0.0800 \, \\(\text{/KWh} = 22.4 \, \\)\]
Adding the purchase price:
\[17 \, \\( + 22.4 \, \\) = 39.4 \, \\(\]
In contrast, a conventional bulb costs \)0.420 for each unit, and \(80 for energy use:
  • Replacing it over 10,000 hours requires 14 bulbs,
    \[14 \, \times 0.420 \, \\) = 5.88 \, \\(\] for bulbs.
  • Adding energy use,
    \[80 \, \\) + 5.88 \, \\( = 85.88 \, \\)\]
The total saving by using an energy-efficient bulb is:
\[85.88 \, \\( - 39.4 \, \\) = 46.48 \, \$\]
These savings add up over time, highlighting the economic benefits of energy-efficient bulbs.
Lifespan comparison
Another important consideration is how long each type of bulb lasts. This directly affects the cost efficiency. Energy-efficient bulbs have a remarkable lifespan of 10,000 hours, compared to traditional bulbs that will only last 750 hours. This makes them far superior in terms of durability.
  • This means fewer bulb replacements, saving on purchase costs.
  • The sustained performance over a longer period also ensures fewer interruptions.
  • It reduces the hassle of frequent changes, especially in locations with hard-to-reach fixtures.
Thus, the longevity of energy-efficient bulbs not only boosts economic savings but also contributes to convenience and practical benefits. Choosing energy-efficient bulbs lessens the frequency of replacement, which is a compelling reason for their adoption.

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Most popular questions from this chapter

If it takes \(4.00 \mathrm{J}\) of work to stretch a Hooke's-law spring \(10.0 \mathrm{cm}\) from its unstressed length, determine the extra work required to stretch it an additional \(10.0 \mathrm{cm} .\)

A 100 -g bullet is fired from a rifle having a barrel \(0.600 \mathrm{m}\) long. Assuming the origin is placed where the bullet begins to move, the force (in newtons) exerted by the expanding gas on the bullet is \(15000+10000 x-25000 x^{2}\) where \(x\) is in meters. (a) Determine the work done by the gas on the bullet as the bullet travels the length of the barrel. (b) What If? If the barrel is 1.00 m long, how much work is done, and how does this value compare to the work calculated in (a)?

Energy is conventionally measured in Calories as well as in joules. One Calorie in nutrition is one kilocalorie, defined as \(1 \mathrm{kcal}=4186 \mathrm{J} .\) Metabolizing one gram of fat can release 9.00 kcal. A student decides to try to lose weight by exercising. She plans to run up and down the stairs in a football stadium as fast as she can and as many times as necessary. Is this in itself a practical way to lose weight? To evaluate the program, suppose she runs up a flight of 80 steps, each \(0.150 \mathrm{m}\) high, in \(65.0 \mathrm{s}\). For simplicity, ignore the energy she uses in coming down (which is small). Assume that a typical efficiency for human muscles is \(20.0 \%\) This means that when your body converts \(100 \mathrm{J}\) from metabolizing fat, \(20 \mathrm{J}\) goes into doing mechanical work (here, climbing stairs). The remainder goes into extra internal energy. Assume the student's mass is \(50.0 \mathrm{kg} .\) (a) How many times must she run the flight of stairs to lose one pound of fat? (b) What is her average power output, in watts and in horsepower, as she is running up the stairs?

While running, a person dissipates about \(0.600 \mathrm{J}\) of mechanical energy per step per kilogram of body mass. If a 60.0 -kg runner dissipates a power of \(70.0 \mathrm{W}\) during a race, how fast is the person running? Assume a running step is \(1.50 \mathrm{m}\) long.

When a 4.00 -kg object is hung vertically on a certain light spring that obeys Hooke's law, the spring stretches \(2.50 \mathrm{cm} .\) If the \(4.00-\mathrm{kg}\) object is removed, (a) how far will the spring stretch if a \(1.50-\mathrm{kg}\) block is hung on it, and (b) how much work must an external agent do to stretch the same spring \(4.00 \mathrm{cm}\) from its unstretched position?
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