WEBVTT
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Hey everyone, welcome back to Office Hours.
00:00:20.640 --> 00:00:27.280
I usually use these episodes to explain the math behind the study hall problems without working through the exact questions.
00:00:27.440 --> 00:00:29.039
Today I want to make an exception.
00:00:29.199 --> 00:00:34.640
Problems 9 and 10 are worth solving together because they test more than the rules of scientific notation.
00:00:34.960 --> 00:00:44.079
They test whether you can read a paragraph, figure out what the question is really asking, and decide what to multiply or divide before you start calculating.
00:00:44.320 --> 00:00:45.359
So here's the plan.
00:00:45.520 --> 00:00:49.759
First, we'll review the small set of scientific notation skills that you need.
00:00:50.000 --> 00:00:55.520
Then we'll work through problems 9 and 10 from beginning to end on the study hall sheet.
00:00:55.920 --> 00:01:00.880
If you haven't opened the study hall sheet yet, that's completely fine.
00:01:00.960 --> 00:01:03.759
I'll give you everything you need right here.
00:01:09.920 --> 00:01:13.840
In the main episode, we followed a simple but remarkable idea.
00:01:14.079 --> 00:01:20.959
Benjamin Franklin poured a teaspoon of oil onto a pond and watched it spread across a large area.
00:01:21.359 --> 00:01:27.920
Much later, Agnes Puckles developed more controlled ways to study very thin films on water.
00:01:28.159 --> 00:01:30.159
The basic question was this.
00:01:30.480 --> 00:01:38.480
If a known amount of oil spreads into an extremely thin layer, what can that tell us about the size of a molecule?
00:01:38.719 --> 00:01:44.719
The question starts with an ordinary amount of oil and ends with the measurements at the molecular level.
00:01:44.959 --> 00:01:50.799
The numbers cover such a wide range that ordinary decimal notation becomes difficult to use.
00:01:51.040 --> 00:01:59.599
Scientific notation gives us a clean way to write these numbers, and, more importantly, a reliable way to calculate with them.
00:02:00.799 --> 00:02:04.480
For everyday quantities, regular notation works well.
00:02:04.719 --> 00:02:12.879
12 eggs, 20 miles,$500, but becomes harder to read when the number is extremely large or extremely small.
00:02:13.120 --> 00:02:17.840
The average distance from Earth to the Sun is about 150 billion meters.
00:02:18.080 --> 00:02:22.000
A red blood cell is about 7 millionths of a meter across.
00:02:22.159 --> 00:02:27.520
You can write both numbers that way, but counting zeros is slow and easy to get wrong.
00:02:27.759 --> 00:02:39.520
Scientific notation rewrites each number in the same basic form, a number from 1 up to, but not including 10, multiplied by a power of 10.
00:02:39.840 --> 00:02:44.960
So 150 billion becomes 1.5 times 10 to the 11th.
00:02:45.120 --> 00:02:49.439
And 7 millionth becomes 7 times 10 to the negative 6.
00:02:49.840 --> 00:02:52.479
The first number is called the coefficient.
00:02:52.639 --> 00:02:55.120
The exponent tells you the scale of the number.
00:02:55.280 --> 00:02:59.199
A positive exponent gives you a number larger than 1.
00:02:59.360 --> 00:03:02.960
A negative exponent gives you a decimal between 0 and 1.
00:03:03.120 --> 00:03:07.360
And that's the whole concept, coefficient times a power of 10.
00:03:07.599 --> 00:03:10.400
Let's go over the sign of the exponents.
00:03:10.719 --> 00:03:15.280
10 to the third means 10 multiplied by itself three times.
00:03:15.520 --> 00:03:16.960
That's 1000.
00:03:17.360 --> 00:03:20.879
So a positive exponent makes the power of 10 larger.
00:03:21.280 --> 00:03:25.599
10 to the negative third means 1 divided by 10 to the third.
00:03:25.759 --> 00:03:30.000
That's 1 1000 or 0.001.
00:03:30.159 --> 00:03:33.680
The negative sign doesn't make the measurement itself negative.
00:03:33.759 --> 00:03:36.719
It tells you that the power of 10 is a fraction.
00:03:36.960 --> 00:03:44.400
When you convert a large number to scientific notation, you move the decimal to the left until the coefficient is between 1 and 10.
00:03:44.639 --> 00:03:48.240
The number of places you moved becomes a positive exponent.
00:03:48.479 --> 00:03:54.560
When you convert a small decimal, you move the decimal to the right until the coefficient is between 1 and 10.
00:03:54.800 --> 00:03:57.840
The number of places you move becomes a negative exponent.
00:03:58.080 --> 00:04:03.039
For example, 82,000 becomes 8.2 times 10 to the fourth.
00:04:03.199 --> 00:04:12.800
The decimal moved four places to the left, and 0.00082 becomes 8.2 times 10 to the negative fourth.
00:04:12.879 --> 00:04:15.520
The decimal moved four places to the right.
00:04:15.759 --> 00:04:18.079
Same digits, just a different scale.
00:04:18.240 --> 00:04:24.560
One of the best reasons to use scientific notation is that exponents give you a quick estimate of size.
00:04:24.720 --> 00:04:31.600
Suppose one value is around 10 to the third, and another is around 10 to the negative sixth.
00:04:31.839 --> 00:04:37.600
Their exponents are nine apart, so the two values are separated by roughly nine powers of 10.
00:04:37.839 --> 00:04:40.959
You can see that before doing any detailed arithmetic.
00:04:41.199 --> 00:04:43.680
This gives you a fast way to check an answer.
00:04:43.839 --> 00:04:52.319
If you expect something close to 10 to the negative ninth, and your calculator gives you 10 to the positive ninth, at that point you're not going to want to move on.
00:04:52.480 --> 00:04:56.560
The sign or one of the exponent operations is probably wrong.
00:04:56.879 --> 00:04:59.839
A calculator will carry out the keys you press.
00:05:00.079 --> 00:05:03.519
It won't tell you whether the setup makes sense.
00:05:03.680 --> 00:05:09.600
The exponent gives you a way to catch a bad setup or a typing error before you trust the result.
00:05:10.160 --> 00:05:12.000
There are two main rules.
00:05:12.240 --> 00:05:18.399
When you multiply numbers in scientific notation, multiply the coefficients and add the exponents.
00:05:18.639 --> 00:05:22.879
When you divide, divide the coefficients and subtract the exponents.
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For example, two times ten to the fourth multiplied by three times ten to the negative second gives six times ten to the second.
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Two times three is six, four plus negative two is two.
00:05:38.639 --> 00:05:40.879
After you calculate, check the coefficient.
00:05:41.279 --> 00:05:46.000
In proper scientific notation, it must be at least one but less than ten.
00:05:46.879 --> 00:05:50.560
Suppose your calculation gives sixteen times ten to the fifth.
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Sixteen is too large for the coefficient, rewrite sixteen as one point six times ten.
00:05:56.639 --> 00:06:00.720
That turns the answer into one point six times ten to the sixth.
00:06:01.759 --> 00:06:04.079
The same issue can happen in the other direction.
00:06:04.319 --> 00:06:13.680
If you get 0.63 times ten to the twenty-fourth, move the decimal one place to the right to make the coefficient six point three.
00:06:13.920 --> 00:06:19.279
Because you made the coefficient ten times larger, you make the power of ten ten times smaller.
00:06:19.519 --> 00:06:22.879
The answer becomes six point three times ten to the twenty-third.
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That last adjustment is easy to skip.
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So build in a final question after every calculation.
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Is my coefficient between one and ten?
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Now we get to the most important part.
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In a multi-step word problem, the exponent rules are usually not the hardest part.
00:06:40.079 --> 00:06:45.040
The hardest part is deciding what the number represents and how they fit together.
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I use three questions.
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First, what am I given?
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Write down each quantity with its unit.
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Second, what am I being asked to find?
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Pay close attention to the target unit.
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Third, what operation changes the units I have into the units I need?
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The units are not an afterthought.
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They often tell you the operation.
00:07:08.319 --> 00:07:11.839
Meters times square meters gives you cubic meters.
00:07:12.000 --> 00:07:18.399
A total volume divided by a volume per molecule gives a number of molecules.
00:07:18.560 --> 00:07:23.040
Once the units line up, then do the exponent arithmetic.
00:07:23.279 --> 00:07:26.959
Let's use that method on two study hall problems.
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So let's set up problem number nine on the study hall problem sheet.
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Here's the information we're given.
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Franklin's teaspoon held about five times ten to the negative six cubic meters of oil.
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Each molecule is about 2.5 times 10 to the negative ninth meters long.
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Its footprint is about 2 times 10 to the negative 19th square meters.
00:07:51.839 --> 00:07:57.040
The question asks approximately how many molecules were in the teaspoon.
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Start with the target.
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We need a count of molecules.
00:08:01.920 --> 00:08:10.319
To find a count, we can divide the total volume of oil by the volume occupied by one molecule.
00:08:10.560 --> 00:08:12.959
We already know the total volume.
00:08:13.279 --> 00:08:18.800
We don't yet have the volume of one molecule, but we have enough information to estimate it.
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A molecule has a length measured in meters and a footprint measured in square meters.
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Multiply them and the units become cubic meters.
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That gives us an estimated volume per molecule.
00:08:32.799 --> 00:08:47.360
So to think about the volume of one molecule, take two times two point five times ten to the negative ninth meters and multiply that by two times ten to the negative meters squared.
00:08:48.240 --> 00:08:53.919
That equals five times ten to the negative twenty-eighth meter cubed.
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For the coefficients, two point five times two is five.
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And for the exponents, negative nine plus negative nineteen is negative twenty-eight.
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So one molecule occupies an estimated five times ten to the negative twenty-eighth cubic meters.
00:09:15.519 --> 00:09:19.600
Now divide the total volume by the volume per molecule.
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Number of molecules is five times ten to the negative six meters cubed divided by five times negative ten to the negative twenty-eighth meters cubed per molecule.
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That's one times ten to the twenty-second molecules.
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Five divided by five is one.
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For the exponents, negative six minus negative twenty-eight becomes negative six plus twenty-eight, which is positive twenty-two.
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The cubic meter units cancel, leaving a count of molecules.
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Our estimate is one times ten to the twenty-second molecules.
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Now do a quick reasonableness check.
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A teaspoon is small to us, but a molecule is extraordinarily small.
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A very large positive exponent is exactly what you would expect when we count how many molecules fit into a teaspoon.
00:10:15.360 --> 00:10:19.200
An answer such as 10 to the fourth would be far too small.
00:10:19.440 --> 00:10:25.519
An answer with a negative exponent would make no sense for the number of molecules in a visible amount of oil.
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Now let's look at problem number 10 on the study hall study sheet.
00:10:30.639 --> 00:10:35.279
Problem 10 uses the same model but changes the amount of oil.
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One mole of the oil occupies about 3.15 times 10 to the negative fourth cubic meters.
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The estimated dimensions of one molecule are unchanged.
00:10:48.240 --> 00:10:58.399
The problem says that the number of molecules can be written as 6.3 times 10 to the k power, and it use it wants us to find k.
00:10:59.039 --> 00:11:04.159
The letter k can make this look like a different kind of problem, but the process is the same.
00:11:04.480 --> 00:11:12.480
We calculate the number of molecules, write the result in proper scientific notation, and then read the exponent.
00:11:12.720 --> 00:11:20.960
From problem 9, we already found the estimated volume of one molecule, 5 times 10 to the negative 28th cubic meters.
00:11:21.200 --> 00:11:25.679
Divide the volume of one mole by the volume of one molecule.
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So the number of molecules in one mole is 3.15 times 10 to the negative 4th meters cubed divided by 5 times 10 to the negative 28th meters cubed per molecule.
00:11:39.600 --> 00:11:44.799
And that equals 0.63 times 10 to the 24th molecules.
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3.15 divided by 5 is 0.63.
00:11:51.120 --> 00:11:56.879
For the exponents, negative 4 minus negative 28 is positive 24.
00:11:57.360 --> 00:12:02.159
That gives us 0.63 times 10 to the 24th.
00:12:02.480 --> 00:12:03.519
But we're not finished.
00:12:03.759 --> 00:12:11.759
0.63 is smaller than 1, so it's not a valid coefficient in standard scientific notation.
00:12:12.240 --> 00:12:18.639
Move the decimal one place to the right from 0.63 to 6.3.
00:12:19.120 --> 00:12:24.399
To keep the value the same, reduce the exponents by 1 from 24 to 23.
00:12:24.879 --> 00:12:33.600
So in standard scientific notation, 0.63 times 10 to the 24th is actually 6.3 times 10 to the 23rd.
00:12:34.080 --> 00:12:37.039
Therefore, k equals 23.
00:12:37.519 --> 00:12:38.480
That's the answer.
00:12:38.639 --> 00:12:40.720
We don't need to write out all the zeros.
00:12:40.879 --> 00:12:46.399
The question asks for the exponent, and the scientific notation already shows it to us.
00:12:47.200 --> 00:12:49.039
There's also a useful check here.
00:12:49.200 --> 00:12:56.960
The accepted value of Avogadro's number is about 6.022 times 10 to the 23rd particles per mole.
00:12:57.120 --> 00:13:02.240
Our simplified estimate gives 6.3 times 10 to the 23rd.
00:13:02.480 --> 00:13:06.639
Those values are close in scale and fairly close in coefficient.
00:13:06.879 --> 00:13:15.039
We should not expect an exact match because our model treats a molecule as a simple volume based on its length and footprint.
00:13:15.279 --> 00:13:20.960
Real molecules are not neat rectangular blocks, and the measurements are approximate.
00:13:21.120 --> 00:13:26.720
But landing near Avogadro's number tells us that the method produces a sensible estimate.
00:13:26.960 --> 00:13:30.879
It also shows why the oil film story matters.
00:13:31.120 --> 00:13:40.159
A visible amount of oil spread across a measurable area can give us information about objects far too small to see directly.
00:13:40.480 --> 00:13:48.960
Scientific notation is what allows the macroscopic measurements and the molecular measurements to stay in the same calculation.
00:13:49.600 --> 00:13:55.759
Let's finish this office hours with a method you can use on any scientific notation word problem.
00:13:56.320 --> 00:13:59.679
Write down what you're given, including the units.
00:13:59.919 --> 00:14:07.519
Write down what you need to find, including its unit, use the units to decide what to multiply or divide.
00:14:07.759 --> 00:14:09.919
Then apply the exponent rules.
00:14:10.159 --> 00:14:13.440
Add when you multiply and subtract when you divide.
00:14:13.679 --> 00:14:21.519
Put the answer into standard scientific notation with a coefficient from 1 up to but not including 10.
00:14:21.759 --> 00:14:26.320
Finally check the exponent and ask whether the scale makes sense.
00:14:26.559 --> 00:14:29.519
The calculation itself may only take a few lines.
00:14:29.679 --> 00:14:33.840
The real skill is building those lines correctly from the information in the problem.
00:14:34.559 --> 00:14:39.759
If you haven't worked through the problems one through eight yet, try them with this same routine.
00:14:39.840 --> 00:14:44.320
Then return to 9 and 10 and see whether the setup feels more manageable.
00:14:44.720 --> 00:14:47.120
Well that's it for this episode of Office Hours.
00:14:47.200 --> 00:14:48.559
I'm Behe Rabani.
00:14:48.879 --> 00:14:54.399
I hope you enjoyed this episode, and stay tuned for the next episode of Story Problem.
00:14:54.799 --> 00:14:55.840
Thank you so much.
00:14:56.000 --> 00:14:57.120
We'll see you next time.
00:00:17.519 --> 00:00:20.320
Hey everyone, welcome back to Office Hours.
00:00:20.640 --> 00:00:27.280
I usually use these episodes to explain the math behind the study hall problems without working through the exact questions.
00:00:27.440 --> 00:00:29.039
Today I want to make an exception.
00:00:29.199 --> 00:00:34.640
Problems 9 and 10 are worth solving together because they test more than the rules of scientific notation.
00:00:34.960 --> 00:00:44.079
They test whether you can read a paragraph, figure out what the question is really asking, and decide what to multiply or divide before you start calculating.
00:00:44.320 --> 00:00:45.359
So here's the plan.
00:00:45.520 --> 00:00:49.759
First, we'll review the small set of scientific notation skills that you need.
00:00:50.000 --> 00:00:55.520
Then we'll work through problems 9 and 10 from beginning to end on the study hall sheet.
00:00:55.920 --> 00:01:00.880
If you haven't opened the study hall sheet yet, that's completely fine.
00:01:00.960 --> 00:01:03.759
I'll give you everything you need right here.
00:01:09.920 --> 00:01:13.840
In the main episode, we followed a simple but remarkable idea.
00:01:14.079 --> 00:01:20.959
Benjamin Franklin poured a teaspoon of oil onto a pond and watched it spread across a large area.
00:01:21.359 --> 00:01:27.920
Much later, Agnes Puckles developed more controlled ways to study very thin films on water.
00:01:28.159 --> 00:01:30.159
The basic question was this.
00:01:30.480 --> 00:01:38.480
If a known amount of oil spreads into an extremely thin layer, what can that tell us about the size of a molecule?
00:01:38.719 --> 00:01:44.719
The question starts with an ordinary amount of oil and ends with the measurements at the molecular level.
00:01:44.959 --> 00:01:50.799
The numbers cover such a wide range that ordinary decimal notation becomes difficult to use.
00:01:51.040 --> 00:01:59.599
Scientific notation gives us a clean way to write these numbers, and, more importantly, a reliable way to calculate with them.
00:02:00.799 --> 00:02:04.480
For everyday quantities, regular notation works well.
00:02:04.719 --> 00:02:12.879
12 eggs, 20 miles,$500, but becomes harder to read when the number is extremely large or extremely small.
00:02:13.120 --> 00:02:17.840
The average distance from Earth to the Sun is about 150 billion meters.
00:02:18.080 --> 00:02:22.000
A red blood cell is about 7 millionths of a meter across.
00:02:22.159 --> 00:02:27.520
You can write both numbers that way, but counting zeros is slow and easy to get wrong.
00:02:27.759 --> 00:02:39.520
Scientific notation rewrites each number in the same basic form, a number from 1 up to, but not including 10, multiplied by a power of 10.
00:02:39.840 --> 00:02:44.960
So 150 billion becomes 1.5 times 10 to the 11th.
00:02:45.120 --> 00:02:49.439
And 7 millionth becomes 7 times 10 to the negative 6.
00:02:49.840 --> 00:02:52.479
The first number is called the coefficient.
00:02:52.639 --> 00:02:55.120
The exponent tells you the scale of the number.
00:02:55.280 --> 00:02:59.199
A positive exponent gives you a number larger than 1.
00:02:59.360 --> 00:03:02.960
A negative exponent gives you a decimal between 0 and 1.
00:03:03.120 --> 00:03:07.360
And that's the whole concept, coefficient times a power of 10.
00:03:07.599 --> 00:03:10.400
Let's go over the sign of the exponents.
00:03:10.719 --> 00:03:15.280
10 to the third means 10 multiplied by itself three times.
00:03:15.520 --> 00:03:16.960
That's 1000.
00:03:17.360 --> 00:03:20.879
So a positive exponent makes the power of 10 larger.
00:03:21.280 --> 00:03:25.599
10 to the negative third means 1 divided by 10 to the third.
00:03:25.759 --> 00:03:30.000
That's 1 1000 or 0.001.
00:03:30.159 --> 00:03:33.680
The negative sign doesn't make the measurement itself negative.
00:03:33.759 --> 00:03:36.719
It tells you that the power of 10 is a fraction.
00:03:36.960 --> 00:03:44.400
When you convert a large number to scientific notation, you move the decimal to the left until the coefficient is between 1 and 10.
00:03:44.639 --> 00:03:48.240
The number of places you moved becomes a positive exponent.
00:03:48.479 --> 00:03:54.560
When you convert a small decimal, you move the decimal to the right until the coefficient is between 1 and 10.
00:03:54.800 --> 00:03:57.840
The number of places you move becomes a negative exponent.
00:03:58.080 --> 00:04:03.039
For example, 82,000 becomes 8.2 times 10 to the fourth.
00:04:03.199 --> 00:04:12.800
The decimal moved four places to the left, and 0.00082 becomes 8.2 times 10 to the negative fourth.
00:04:12.879 --> 00:04:15.520
The decimal moved four places to the right.
00:04:15.759 --> 00:04:18.079
Same digits, just a different scale.
00:04:18.240 --> 00:04:24.560
One of the best reasons to use scientific notation is that exponents give you a quick estimate of size.
00:04:24.720 --> 00:04:31.600
Suppose one value is around 10 to the third, and another is around 10 to the negative sixth.
00:04:31.839 --> 00:04:37.600
Their exponents are nine apart, so the two values are separated by roughly nine powers of 10.
00:04:37.839 --> 00:04:40.959
You can see that before doing any detailed arithmetic.
00:04:41.199 --> 00:04:43.680
This gives you a fast way to check an answer.
00:04:43.839 --> 00:04:52.319
If you expect something close to 10 to the negative ninth, and your calculator gives you 10 to the positive ninth, at that point you're not going to want to move on.
00:04:52.480 --> 00:04:56.560
The sign or one of the exponent operations is probably wrong.
00:04:56.879 --> 00:04:59.839
A calculator will carry out the keys you press.
00:05:00.079 --> 00:05:03.519
It won't tell you whether the setup makes sense.
00:05:03.680 --> 00:05:09.600
The exponent gives you a way to catch a bad setup or a typing error before you trust the result.
00:05:10.160 --> 00:05:12.000
There are two main rules.
00:05:12.240 --> 00:05:18.399
When you multiply numbers in scientific notation, multiply the coefficients and add the exponents.
00:05:18.639 --> 00:05:22.879
When you divide, divide the coefficients and subtract the exponents.
00:05:23.360 --> 00:05:32.959
For example, two times ten to the fourth multiplied by three times ten to the negative second gives six times ten to the second.
00:05:33.439 --> 00:05:37.920
Two times three is six, four plus negative two is two.
00:05:38.639 --> 00:05:40.879
After you calculate, check the coefficient.
00:05:41.279 --> 00:05:46.000
In proper scientific notation, it must be at least one but less than ten.
00:05:46.879 --> 00:05:50.560
Suppose your calculation gives sixteen times ten to the fifth.
00:05:50.959 --> 00:05:55.920
Sixteen is too large for the coefficient, rewrite sixteen as one point six times ten.
00:05:56.639 --> 00:06:00.720
That turns the answer into one point six times ten to the sixth.
00:06:01.759 --> 00:06:04.079
The same issue can happen in the other direction.
00:06:04.319 --> 00:06:13.680
If you get 0.63 times ten to the twenty-fourth, move the decimal one place to the right to make the coefficient six point three.
00:06:13.920 --> 00:06:19.279
Because you made the coefficient ten times larger, you make the power of ten ten times smaller.
00:06:19.519 --> 00:06:22.879
The answer becomes six point three times ten to the twenty-third.
00:06:23.279 --> 00:06:25.759
That last adjustment is easy to skip.
00:06:25.920 --> 00:06:29.120
So build in a final question after every calculation.
00:06:29.279 --> 00:06:31.600
Is my coefficient between one and ten?
00:06:32.639 --> 00:06:34.639
Now we get to the most important part.
00:06:35.040 --> 00:06:39.920
In a multi-step word problem, the exponent rules are usually not the hardest part.
00:06:40.079 --> 00:06:45.040
The hardest part is deciding what the number represents and how they fit together.
00:06:45.360 --> 00:06:46.959
I use three questions.
00:06:47.199 --> 00:06:48.879
First, what am I given?
00:06:49.040 --> 00:06:51.839
Write down each quantity with its unit.
00:06:52.160 --> 00:06:54.800
Second, what am I being asked to find?
00:06:55.040 --> 00:06:57.920
Pay close attention to the target unit.
00:06:58.160 --> 00:07:03.279
Third, what operation changes the units I have into the units I need?
00:07:03.519 --> 00:07:05.600
The units are not an afterthought.
00:07:05.759 --> 00:07:07.839
They often tell you the operation.
00:07:08.319 --> 00:07:11.839
Meters times square meters gives you cubic meters.
00:07:12.000 --> 00:07:18.399
A total volume divided by a volume per molecule gives a number of molecules.
00:07:18.560 --> 00:07:23.040
Once the units line up, then do the exponent arithmetic.
00:07:23.279 --> 00:07:26.959
Let's use that method on two study hall problems.
00:07:27.199 --> 00:07:31.199
So let's set up problem number nine on the study hall problem sheet.
00:07:31.439 --> 00:07:33.199
Here's the information we're given.
00:07:33.439 --> 00:07:39.680
Franklin's teaspoon held about five times ten to the negative six cubic meters of oil.
00:07:39.920 --> 00:07:45.680
Each molecule is about 2.5 times 10 to the negative ninth meters long.
00:07:46.000 --> 00:07:51.439
Its footprint is about 2 times 10 to the negative 19th square meters.
00:07:51.839 --> 00:07:57.040
The question asks approximately how many molecules were in the teaspoon.
00:07:57.439 --> 00:07:59.120
Start with the target.
00:07:59.360 --> 00:08:01.680
We need a count of molecules.
00:08:01.920 --> 00:08:10.319
To find a count, we can divide the total volume of oil by the volume occupied by one molecule.
00:08:10.560 --> 00:08:12.959
We already know the total volume.
00:08:13.279 --> 00:08:18.800
We don't yet have the volume of one molecule, but we have enough information to estimate it.
00:08:19.040 --> 00:08:24.959
A molecule has a length measured in meters and a footprint measured in square meters.
00:08:25.279 --> 00:08:28.639
Multiply them and the units become cubic meters.
00:08:28.879 --> 00:08:32.559
That gives us an estimated volume per molecule.
00:08:32.799 --> 00:08:47.360
So to think about the volume of one molecule, take two times two point five times ten to the negative ninth meters and multiply that by two times ten to the negative meters squared.
00:08:48.240 --> 00:08:53.919
That equals five times ten to the negative twenty-eighth meter cubed.
00:08:54.159 --> 00:08:59.440
For the coefficients, two point five times two is five.
00:08:59.759 --> 00:09:07.039
And for the exponents, negative nine plus negative nineteen is negative twenty-eight.
00:09:07.679 --> 00:09:14.720
So one molecule occupies an estimated five times ten to the negative twenty-eighth cubic meters.
00:09:15.519 --> 00:09:19.600
Now divide the total volume by the volume per molecule.
00:09:19.919 --> 00:09:32.480
Number of molecules is five times ten to the negative six meters cubed divided by five times negative ten to the negative twenty-eighth meters cubed per molecule.
00:09:32.720 --> 00:09:36.480
That's one times ten to the twenty-second molecules.
00:09:36.799 --> 00:09:38.879
Five divided by five is one.
00:09:39.360 --> 00:09:49.679
For the exponents, negative six minus negative twenty-eight becomes negative six plus twenty-eight, which is positive twenty-two.
00:09:49.919 --> 00:09:54.159
The cubic meter units cancel, leaving a count of molecules.
00:09:54.480 --> 00:09:59.600
Our estimate is one times ten to the twenty-second molecules.
00:10:00.000 --> 00:10:02.320
Now do a quick reasonableness check.
00:10:02.480 --> 00:10:06.960
A teaspoon is small to us, but a molecule is extraordinarily small.
00:10:07.279 --> 00:10:15.039
A very large positive exponent is exactly what you would expect when we count how many molecules fit into a teaspoon.
00:10:15.360 --> 00:10:19.200
An answer such as 10 to the fourth would be far too small.
00:10:19.440 --> 00:10:25.519
An answer with a negative exponent would make no sense for the number of molecules in a visible amount of oil.
00:10:25.759 --> 00:10:30.320
Now let's look at problem number 10 on the study hall study sheet.
00:10:30.639 --> 00:10:35.279
Problem 10 uses the same model but changes the amount of oil.
00:10:35.519 --> 00:10:43.360
One mole of the oil occupies about 3.15 times 10 to the negative fourth cubic meters.
00:10:43.600 --> 00:10:48.000
The estimated dimensions of one molecule are unchanged.
00:10:48.240 --> 00:10:58.399
The problem says that the number of molecules can be written as 6.3 times 10 to the k power, and it use it wants us to find k.
00:10:59.039 --> 00:11:04.159
The letter k can make this look like a different kind of problem, but the process is the same.
00:11:04.480 --> 00:11:12.480
We calculate the number of molecules, write the result in proper scientific notation, and then read the exponent.
00:11:12.720 --> 00:11:20.960
From problem 9, we already found the estimated volume of one molecule, 5 times 10 to the negative 28th cubic meters.
00:11:21.200 --> 00:11:25.679
Divide the volume of one mole by the volume of one molecule.
00:11:26.000 --> 00:11:39.360
So the number of molecules in one mole is 3.15 times 10 to the negative 4th meters cubed divided by 5 times 10 to the negative 28th meters cubed per molecule.
00:11:39.600 --> 00:11:44.799
And that equals 0.63 times 10 to the 24th molecules.
00:11:45.200 --> 00:11:50.639
3.15 divided by 5 is 0.63.
00:11:51.120 --> 00:11:56.879
For the exponents, negative 4 minus negative 28 is positive 24.
00:11:57.360 --> 00:12:02.159
That gives us 0.63 times 10 to the 24th.
00:12:02.480 --> 00:12:03.519
But we're not finished.
00:12:03.759 --> 00:12:11.759
0.63 is smaller than 1, so it's not a valid coefficient in standard scientific notation.
00:12:12.240 --> 00:12:18.639
Move the decimal one place to the right from 0.63 to 6.3.
00:12:19.120 --> 00:12:24.399
To keep the value the same, reduce the exponents by 1 from 24 to 23.
00:12:24.879 --> 00:12:33.600
So in standard scientific notation, 0.63 times 10 to the 24th is actually 6.3 times 10 to the 23rd.
00:12:34.080 --> 00:12:37.039
Therefore, k equals 23.
00:12:37.519 --> 00:12:38.480
That's the answer.
00:12:38.639 --> 00:12:40.720
We don't need to write out all the zeros.
00:12:40.879 --> 00:12:46.399
The question asks for the exponent, and the scientific notation already shows it to us.
00:12:47.200 --> 00:12:49.039
There's also a useful check here.
00:12:49.200 --> 00:12:56.960
The accepted value of Avogadro's number is about 6.022 times 10 to the 23rd particles per mole.
00:12:57.120 --> 00:13:02.240
Our simplified estimate gives 6.3 times 10 to the 23rd.
00:13:02.480 --> 00:13:06.639
Those values are close in scale and fairly close in coefficient.
00:13:06.879 --> 00:13:15.039
We should not expect an exact match because our model treats a molecule as a simple volume based on its length and footprint.
00:13:15.279 --> 00:13:20.960
Real molecules are not neat rectangular blocks, and the measurements are approximate.
00:13:21.120 --> 00:13:26.720
But landing near Avogadro's number tells us that the method produces a sensible estimate.
00:13:26.960 --> 00:13:30.879
It also shows why the oil film story matters.
00:13:31.120 --> 00:13:40.159
A visible amount of oil spread across a measurable area can give us information about objects far too small to see directly.
00:13:40.480 --> 00:13:48.960
Scientific notation is what allows the macroscopic measurements and the molecular measurements to stay in the same calculation.
00:13:49.600 --> 00:13:55.759
Let's finish this office hours with a method you can use on any scientific notation word problem.
00:13:56.320 --> 00:13:59.679
Write down what you're given, including the units.
00:13:59.919 --> 00:14:07.519
Write down what you need to find, including its unit, use the units to decide what to multiply or divide.
00:14:07.759 --> 00:14:09.919
Then apply the exponent rules.
00:14:10.159 --> 00:14:13.440
Add when you multiply and subtract when you divide.
00:14:13.679 --> 00:14:21.519
Put the answer into standard scientific notation with a coefficient from 1 up to but not including 10.
00:14:21.759 --> 00:14:26.320
Finally check the exponent and ask whether the scale makes sense.
00:14:26.559 --> 00:14:29.519
The calculation itself may only take a few lines.
00:14:29.679 --> 00:14:33.840
The real skill is building those lines correctly from the information in the problem.
00:14:34.559 --> 00:14:39.759
If you haven't worked through the problems one through eight yet, try them with this same routine.
00:14:39.840 --> 00:14:44.320
Then return to 9 and 10 and see whether the setup feels more manageable.
00:14:44.720 --> 00:14:47.120
Well that's it for this episode of Office Hours.
00:14:47.200 --> 00:14:48.559
I'm Behe Rabani.
00:14:48.879 --> 00:14:54.399
I hope you enjoyed this episode, and stay tuned for the next episode of Story Problem.
00:14:54.799 --> 00:14:55.840
Thank you so much.
00:14:56.000 --> 00:14:57.120
We'll see you next time.
