In this lesson, we're going to work on some dimensional analysis problems. So let's begin with this one. How many seconds are there in a year? So what we're going to do is we're going to convert from years to days and then days to hours. Hours to minutes and minutes to seconds.
So let's start with what we're given. We're given one year and we want to convert that into seconds. So what is the conversion factor that's going to take us from years to days? We know on average there's 365 days per year, except if you're dealing with a leap year, which there's 366. Well, technically, if you average it, it's 365.25.
But for problems like this, if you put 365 days, you'll be okay. Now let's convert days to hours. There's 24 hours in a day.
So we can cross out the unit days. And now let's convert hours to minutes. There's 60 minutes in an hour. and 60 seconds in a minute. So notice how we set up the conversion fractions.
We do it in such a way that every unit will cancel except the desired unit, which is seconds. That's the only unit that we should have at the end of this problem. To get the answer, we need to multiply by all of the numbers on the numerators of the fractions.
So it's going to be 1 times 365 times 24 times 60 times 60. The answer is going to be 31,536,000 seconds. So that's how many seconds there are in a year that's defined as 365 days. Now, if you choose 365.25, your answer will change slightly.
It would be 31,557,600 seconds. Let's move on to the next problem. A car is traveling at 45 miles per hour.
How fast is it going in meters per second? So how can we convert from miles per hour to meters per second? So this is a two-part problem.
First, we need to convert the units of length, miles to meters. What we could do is convert miles to kilometers, and then kilometers to meters. Next, we need to convert the units of time from hours to seconds, which we already know how to do.
We can convert from hours to minutes, and then minutes to seconds. Now it's helpful to write the conversion factors that you're going to use. 1 mile is equal to 1.609 kilometers. And 1 kilometer, think of the word kilo.
Kilo is 1,000, so 1 kilometer is 1,000 meters. And we know that 1 hour is equal to 60 minutes. And one minute is equal to 60 seconds. So those are the conversion factors that we're going to use in this problem. Now let's go ahead and get started.
So we have 45 miles per hour. And let's convert miles to kilometers. So we have miles on top.
We want to put the same unit on the bottom. So here's our conversion factor. I'm going to put this part on the bottom of the second fraction.
And then the other part on the top of the second fraction. So the unit's miles will cancel. Now let's use our next conversion factor to convert from kilometers to meters. So since we have kilometers on top, in the second fraction...
I'm going to put kilometers on the bottom of the third fraction. And then the other side of the equation is going to go on top. So now we can cross out kilometers. So we have our desired unit meters, so we can leave that alone. Now let's focus on units of time.
Let's convert hours to seconds. But first we'll convert hours to minutes. So let's use this conversion factor.
Notice we have hours on the bottom. So we need to put it on top of the 4th fraction. So we're going to put this here. The other part of the conversion factor is going to go on the bottom. So now we can cross out hours.
Finally, we can use the last conversion factor to go from minutes to seconds. And now we can cross out the unit minutes. So we're left with meters on top and seconds on the bottom. So we have the speed in meters per second.
So now we need to do the math. Everywhere you see a 1, you can ignore. We're going to multiply by the numbers on top, and then divide by the numbers on the bottom.
Let's begin. It's going to be 45 times 1.609 times 1000, divided by 60, and then divide that result by 60. So you should get 20.1 meters per second. So that's how you can convert from miles per hour to meters per second.
Now let's move on to number three. The density of aluminum metal is 2700 kilograms per cubic meter. What is the density in grams per milliliter? Go ahead and try that one.
So we're given the density in kilograms per cubic meter, and we want to convert it to the density in grams per milliliter. So we need to convert the mass from kilograms to grams. We could do that in a single step.
One kilogram is a thousand grams. And then we need to convert the volume portion of density from cubic meters to milliliters. So what we can do is convert cubic meters to cubic centimeters.
By the way, one meter is a hundred centimeters. And then we can convert cubic centimeters to milliliters. One milliliter. is equivalent to one cubic centimeter. So let's begin.
We're given 2700 kilograms per cubic meter. That's the density of aluminum metal. We could use this one to go from kilograms to grams. So one kilogram is equivalent to a thousand grams.
So now we can cross out the unit kilograms. Now, let's use the next one. Let's go from meters to centimeters.
We know that one meter is equal to 100 centimeters. Since we have meters on the bottom, I decided to put meters on top in a third fraction. But now, this is the part you need to pay special attention to.
Notice that we don't just have meters. We have cubic meters. That's a meter times a meter times a meter.
What we need to do is raise this to the third power, so this becomes cubic meters. If we raise this to the third power, 1 meter raised to the third power, that's 1 meter times 1 meter times 1 meter. That becomes 1 meter cubed.
Now 100 centimeters raised to the third power, that's 100 centimeters times 100 centimeters times 100 centimeters. So that is 1 million cubic centimeters. Or you can write it as 1 times 10 to the 6th.
So basically what you do is... You can take the cube of this equation and you get 1 cubic meter is equal to a million cubic centimeters. So now we can cross out the unit. cubic meters. If you're wondering why this screen looks a little different, I had to do some video editing.
Now let's finish this problem. So right now we can cross out the unit cubic meters. We have grams on top, but we need to get milliliters on the bottom.
So we could use our final conversion factor. 1 cubic centimeter is equal to 1 milliliter. So now we could cancel the unit cubic centimeters.
So what we have is the unit grams on top and the unit milliliters on the bottom. So now we just got to do the math. So we're going to multiply 2700 by 1000 and take that result, divide it by a million, or 1 times 10 to the 6th.
The final answer is going to be 2.7 grams per milliliter. Anytime you need to convert from kilograms per cubic meter to grams per milliliter, simply divide by a thousand. Because in physics, the density of objects are typically reported in kilograms per cubic meter.
meter but in chemistry you'll find that the density is usually reported in grams per milliliter. So this is a common conversion that you may encounter when going between physics and chemistry. Number four, a rectangular doormat has a length of 26 inches and a width of 18 inches. What is the area of the doormat in square inches and square feet? So let's draw a picture.
Let's say this is the doormat. The length is 26 inches. The width is 18 inches. To calculate the area, the area of a rectangle is the length times the width. It's L times W.
So in this example, we need to multiply 26 inches by 18 inches. This is going to be 468. Now we're multiplying inches to the first power times inches to the first power. So when you multiply by a common base, you need to add the exponents. 1 plus 1 is 2. So we get the area in square inches.
So that's the answer to the first part of the question. Now we need to get the area in square feet. So we're going to convert it from square inches to square feet. So how many inches in a foot?
We know that there's 12 inches in one foot, so we have inches on the top left, but we're going to put it on the bottom right. Now notice that it's inches squared, so we need to square. this conversion factor so we can get square feet. So the answer is going to be 468 divided by 12 squared. 12 squared is 144. So you should get 3.25 square feet as your final answer.
So that's how you can convert from square inches to square feet. Number five. John can read 15 pages of a certain book every 45 minutes.
How many hours will it take him to read the entire 200-page book? So think about what we're given. We're given 200 pages, and from that... We need to find out hours. How many hours it will take him to read 200 pages.
So we're converting pages to hours. How can we do this? What conversion factors do we have to go from pages to hours? Well, the first sentence connects pages to minutes.
So we know that he can read 15 pages. every 45 minutes. And we know the conversion from minutes to hours. One hour is 60 minutes. So we have everything that we need.
What we need to do is convert from pages to minutes, and then minutes to hours. So let's start with what we're given, which is 200 pages. And let's use this to convert from pages to minutes.
So in 45 minutes, he can read 15 pages. So we can cancel out the unit pages. And then we'll use this conversion factor to go from minutes to hours.
There's 60 minutes in one hour. So we can cross out minutes, and now we can get the answer. So we're going to multiply by the numbers on top and divide by the numbers on the bottom.
So it's 200 times 45, that's 9,000, divided by 15, that gives us 600, divided by 60, it gives us 10. So it's going to take him 10 hours to read 200 pages of this book.