00:01
So in this problem, we're going to look at how we can see whether an led or an incandescent light bulb is more cost effective for given number of hours and find the break -even point.
00:09
So i don't have the data that was mentioned in this problem, so we're going to make some assumptions here.
00:14
Firstly, that it is electricity is 0 .12 cents or 0 .12 or 0 .12 per kilowatt hour, which is a good average for a country like the u .s.
00:29
For the incandescent bulb, we're going to say that it is 100 watts while that was given, that it has about a thousand hour life, and then that it costs $2.
00:48
And then for the led, we know that it is 20 watts, and we're going to say that it lasts 20 ,000 hours, which is a low estimate, but for this problem it doesn't really matter.
01:04
So we can basically ignore that.
01:08
And let's say it costs $20, which is a bit expensive, but good for an estimate.
01:16
So first, let's look at how we would see a graph of this before we work in numbers.
01:22
We have our plot, and on the x -axis, we're going to have time.
01:27
And we're going to have this in 1 ,000 hours.
01:31
So that 1 here is actually 1 ,000 all in.
01:36
It'll just mean we don't need as many zeros in the math.
01:41
And then on the y -axis, we'll have our cost in dollars.
01:46
So the bulbs both cost something to purchase, so they started some value for the incandescent here.
01:54
And i'm going to abbreviate incandescent.
01:57
And the led, though, is much more expensive.
02:01
And over time, it costs money to run the electricity.
02:03
So these prices should just keep going up continually at some rate.
02:15
And we know that the incandescent uses more power.
02:18
So we would expect the slope to be a bit higher.
02:23
But at some point we need to replace the bulb, so at 1 ,000 hours, and then the price will jump up a bit because we've had to spend money replacing the bulb.
02:37
And then the incandescent will continue growing at the same weight.
02:41
We're going to replace it and so forth.
02:44
And so what we need to do is find where this crossing is.
02:49
So what we're going to do is do this in parts because there's not a clean way to represent this line with discontinuities as one formula.
02:58
Because if it's just two lines, we could just do algebra and find it analytically.
03:03
So here we're going to go through a couple of steps.
03:06
So first, we want to find the slope of these lines, which is how much money they cost in a thousand hours.
03:15
So we know powers build in kilowatt hours.
03:18
So first we need to know that for the incandescent bulb, it uses 100 watts.
03:25
We're considering this over 1 ,000 hours to get our slope here.
03:31
And so we need to do some dimensional analysis where we multiply by factors of things we want over things we have.
03:38
So we want this first in kilowatt hours, and we know that a kilowatt hour is a thousand watt hours.
03:48
And if we see here that the watts cancel, the hours cancel, and then we see here, and the watts cancel, and so we're left with kilowatt hours, what we want.
03:56
And we know that we have 12 cents or 12 cents per kilowatt hour.
04:18
And so once again, the kilowatt hours cancel.
04:27
And so we can put this in our calculator.
04:29
And we see that it costs $12 per 1 ,000 to operate the incandescent light bulb.
04:41
For the led, we're going to do the same exact arithmetic, except we start with the 20 watts.
04:47
And all of this is the same.
04:49
And we work that out and we get $2 .4 per thousand hours.
04:59
So you can see that it's a lot more efficient.
05:02
And of course, these will depend on the specific, like the usages of the bulbs and then the cost of electricity, but the scale would be the same.
05:09
So we see that it's five times more power and so five times more price.
05:14
So now we can start making a table with the incandescent bulb and the led.
05:22
At zero hours, the only thing we've done is buy the bulb.
05:26
So just our, and everything, all this is going to be in dollars.
05:31
So at zero hours, we've just bought the light bulb.
05:35
So $2 for the incandescent, and then we said the led was 20.
05:38
At 1 ,000 hours, we've bought the light bulb, and we've also used 1 ,000 hours with electricity.
05:50
And we see, for the incandescent, it's $12.
05:53
So we're going to add that to our $2 .00.
05:57
We've spent.
05:57
So we, before we replace the bulb, we spent $14.
06:05
Then though, the bulb burns out at a thousand hours, so we need to replace it, spend another $2, and we're going to see $16 there.
06:12
And we'll write both these values just so we can see both of these points, because it could be that the led efficiency crosses somewhere in between, where we need to know where this vertical line is.
06:31
For the led, though, in 1 ,000 hours, we only just burn electricity.
06:35
So we we add our $2 .4 of electricity.
06:38
So we see that right now are $22 .24.
06:41
So it's more expensive than anywhere on this point, so we need to keep going.
06:47
So now we go to 2 ,000 hours...