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.
