Video Transcript
A 1.8-kilowatt kettle has an efficiency of 0.92. It is used to boil 1.3 kilograms of water, which requires 440 kilojoules of energy. How many minutes does it take the kettle to boil the water? Give your answer to the nearest minute.
So in this example, we have a kettle, a device used for heating liquids. Right at the outset, we’re told that this is a 1.8-kilowatt kettle. Now a watt, we can recall, is the unit of power, energy per unit time. So when we’re told that this kettle is a 1.8-kilowatt kettle, that means it can deliver 1.8 thousand watts of power. And since a single watt of power is equivalent to one joule of energy every second, this means that this kettle is capable of delivering 1.8 thousand joules of energy to whatever liquid is in the kettle every second. We could say that this is the heating rate of the kettle.
The slight modification to this, though, is that our kettle isn’t 100 percent efficient. Rather, it has an efficiency of 0.92. That means that not all of this energy, not all of the 1.8 kilojoules every second, actually makes it to the liquid heated in the kettle. Only 92 out of 100 parts of that energy makes it into the liquid. That’s what that efficiency of 0.92 means.
We’re told that this kettle is used to boil water and that the energy requirement for that is 440 kilojoules. We want to know how many minutes does it take the kettle to boil this amount. Now here is one way to think of it. The total energy we need to do this, to boil this water, is 440 kilojoules, or 440000 joules. Every second, our kettle delivers 1.8 kilojoules of energy, of which 92 percent makes it into the liquid. Every second that the kettle is heating the water, 1.8 times 10 to the third multiplied by 0.92 joules of energy goes into the water. That’s the kettle’s heating rate multiplied by its efficiency. Let’s call this modified heating rate, the heating rate that takes into account the efficiency of the kettle, capital 𝑅.
Now that we’ve written down a value for 𝑅, we have the true heating rate of the water in the kettle. This is great because we’re told that the total amount of energy we need to supply is 440 kilojoules. So if we divide this amount by the heating rate, capital 𝑅, then look what happens when we plug in the value for 𝑅. When we do that, we get an overall expression that has units of kilojoules per joule per second.
What we’re going to do next is change these units of kilojoules to joules. We do that by replacing 440 with 440 times 10 to the third. That’s the conversion between kilojoules and joules. Now that we’ve done that, look what happens to these units of joules. We have one unit in the numerator and one in the denominator. That means that they cancel one another out.
And then, look at this. What if we multiply this overall fraction by the fraction, seconds divided by seconds? Algebraically, we can do that and the expression won’t change because we’re just multiplying by what’s effectively one, seconds divided by seconds. But the purpose of doing that, if we look at the lower half of this expression, is to see that those units of seconds now cancel out with one another. If we then remove the units we’ve cancelled out and then tidy up this expression, we find that what remains is an overall number that we can calculate in units of seconds. And this time, by the way, is the total time it would take for this amount of water to boil in the kettle.
We’ve made lots of progress towards our answer, but we noticed that even if we were to calculate this fraction, we wouldn’t quite be there. That’s because we’re asked for how many minutes it takes for the water to boil rather than seconds. To write our answer in these terms, we’ll want to convert the units in this expression, seconds into minutes. To do that, we can recall that one minute is equal to 60 seconds of time. This means that if we multiply this fraction by yet another fraction, which we’ll write as one minute divided by 60 seconds, then once again we’re multiplying by what’s essentially one, since we’ve said that one minute is equal to 60 seconds. But the reason we’re doing this is so that the unit of seconds in numerator and denominator cancel out and we’re left with units of minutes.
See what happened. We had units of seconds in the numerator on this fraction and units of seconds in the denominator on this one. So when we multiply the two fractions together, those two units cancel out. All that remains, then, from a units perspective is this unit of minutes. That’s the unit we wanna give our answer in terms of.
We’re finally ready to calculate this whole expression and write down what we find. 440 times 10 to the third divided by 1.8 times 10 to the third multiplied by 0.92 all multiplied by one divided by 60 gives us approximately 3.75. And the units for this is minutes. But notice, though, that even this isn’t quite our final answer because we’re asked to give our answer to the nearest minute. That means we’ll take this number 3.75 and round it to the nearest minute. We can see that the whole number that 3.75 is closest to is four, so we write that as our answer. To the nearest minute, the time required to boil this water is four minutes.