Question Video: Finding the Total Energy Output of a Device given Its Efficiency and Useful Energy Output | Nagwa Question Video: Finding the Total Energy Output of a Device given Its Efficiency and Useful Energy Output | Nagwa

Question Video: Finding the Total Energy Output of a Device given Its Efficiency and Useful Energy Output Physics

A laptop has an energy efficiency of 0.18. The laptop is used for 1 hour and during this time 88.6 kJ of energy is wasted in the form of heat and sound. How much energy was the laptop supplied with over this time? Give your answer to 3 significant figures.

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Video Transcript

A laptop has an energy efficiency of 0.18. The laptop is used for one hour and during this time 88.6 kilojoules of energy is wasted in the form of heat and sound. How much energy was the laptop supplied with over this time? Give your answer to three significant figures.

In this example, we have a laptop. And we’re told its energy efficiency is 0.18. We’ll represent this efficiency using a shorthand notation. Let’s call it a lowercase 𝑒. In this exercise, we want to know how much energy the laptop was supplied with over a one-hour interval, during which it wasted 88.6 kilojoules of energy. To get started on our solution, let’s recall the general equation for the efficiency of a device. A device’s efficiency is given as the ratio of its output, whether in joules of energy or watts of power or some other unit, to its input, in those same units.

In the case of this laptop, our efficiency is given as 0.18. Which means that 0.18 is equal to the useful output of the laptop divided by its input, in this case energy. Interestingly, in our problem statement, we’re not told either our output, that is a useful power output, or the energy input to the laptop. In fact, that input energy is what we want to solve for. Even though we’re not told these values, we are told how much energy the laptop wastes over this time period. Over one hour, 88.6 kilojoules of energy was wasted due to heat and sound.

Now, this output, the numerator of our efficiency fraction, is related to the input as well as the energy wasted. Written as an equation, we can say that the energy input to the laptop is equal to the useful energy output plus the wasted energy. So we can rearrange this equation to solve for the useful energy output, in terms of the waste and the input. Subtracting the energy wasted from both sides of the equation, we find that the useful energy output from this laptop is equal to the energy input minus the wasted energy. Knowing this, we can then take this term, which is equal to output, and substitute it in for output in our efficiency equation.

When we take a look at this new expression for the laptop efficiency, we see it involves the quantity we want to solve for, the energy input, in terms of quantities we know, the efficiency and the energy waste. To solve for the total energy input, let’s multiply both sides of the equation by that value. When we do, that term cancels out from the right-hand side of the equation. And then, if we add the energy wasted to both sides of the equation and finally subtract 0.18 times the input from both sides, we find that the energy wasted is equal to the energy input multiplied by the quantity one minus 0.18. One minus 0.18 is equal 0.82. And if, as a last step, we divide both sides of the equation by this value, then we see that factor cancels on the right-hand side. And finally, we have an expression for the total energy input, what we want to solve for, in terms of the energy wasted.

At this point, we can use the fact that the energy wasted is 88.6 kilojoules and substitute that value in for waste in our equation. 88.6 kilojoules divided by 0.82 is equal, to three significant figures, to 108 kilojoules. That’s the amount of energy the laptop was supplied with over this one-hour interval.

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