Measurement: Length, Area and Volume Part I

Transcription

1 IDS 101 Name Measurement: Length, Area and Volume Part I If we ask someone the size of a common object, such as a dime or penny, most people come pretty close to the actual size. However, objects that are much larger and much smaller are not as apparent to most of us. For example, how large is an atom? Or the solar system? To better understand the scale of the objects we are going to be discussing this year, let s explore the sizes of various objects. We have a couple of activities to help you understand the sizes of things. Activity #1: It is helpful to convert really large and really small object to a scale that is meaningful to us. First, estimate the distance from your home to the college: Now, convert that distance to kilometers, then to meters, and then to centimeters. (use your previous module on unit conversions for these conversions). For example if you live 10 miles away from the college: 1.6km 1000m 100cm 10milesX X X 1.6X10 1mile 1km 1m Do this same calculation for your distance: 6 cm The distance from the Earth to the Sun is about 150 million kilometers 1. To determine the scale factor for our activity, divide the 150 million kilometer distance to the Sun by the distance to your home in centimeters. In other words, make a mental model such that the distance to the sun is shrunk to 10 miles. Use the 10 mile distance example again: 150,000,000km km ,600,000cm cm This means that each centimeter of distance from GRCC to your home is kilometers. So, the distance from here to Portland (246 km in real distance) would be: 246 km km cm 2.6 cm 1 This is an approximate average distance. The orbit of the Earth is almost circular, but the actual distance varies a little during the year. When is the Earth closest to the Sun? This answer may surprise you!

2 IDS 101 Sizes of things page 2 This means that on our scale, Portland Oregon, would be 2.6 cm (about one inch) away. For the following objects determine the distance from GRCC on your scale: (show an example calculation in the space below) Distance (km) The diameter of the Earth 13,000 Distance on your scale Distance to the Moon 384,000 Distance to Mars 7.5 X 10 7 Distance to Venus 4 X 10 7 Distance to Mercury 7.7 X 10 7 Distance light travels in a 9.46 x year (a light year) Distance to the nearest star 3.99 X Distance to the center of our 2.46 X galaxy Distance to the next galaxy 2.4 x 10 18

3 IDS 101 Sizes of things page 3 Activity #2- Getting smaller We will use the same distance to your home in centimeters, but this time that distance is equivalent to a 26 cm (10 inch) maple leaf. In other words, 26 cm leaf is magnified so that it is represented by a mental model that is 10 miles (1.6 X 10 6 cm) across. Develop a new scale and complete the table below: Diameters Distance (in meters) Distance using your scale Water molecule 1.5 X Sucrose 4.4 X Hemoglobin 3.3 X 10-9 Carbon atom 1.4 X Nucleus of Carbon atom 2.2 X Proton 8.7 X Animal cells 2 X 10-5 Nucleus of an animal cell Single DNA molecule (diameter, not length) Human chromosome (diameter) 5 X X X 10-8 From the list of distances on your scale, rank order the items in this list from largest to smallest:

4 IDS 101 Sizes of things page 4 A summary: What is larger, the nucleus of a cell or the nucleus of an atom? What is larger, an atom or a molecule? What is larger, an atom or a proton? If the Earth was the size of your head (about 18 cm or 7 inches), the Moon would be about the size of a racquetball. How far away from your head would the racquetball be if it was the Moon? Use the racquetballs provided to estimate the distance. You can check your estimation by talking with an instructor. Using this same scale, how large would the sun be? And how far away would the sun be from you? Powers of 10: Imagine cutting a strip of paper so that the remaining piece was only 10% of the original piece of paper. So, if an original strip of paper is 1 meter long, after cutting it would be 1/10 of a meter or 10 centimeters long. To understand this further, let s use the images below. The first image is of Danielle, Kristen, and Juno, a dog who loves balls and Frisbees. The tape distance between Danielle and Kristen is 10 meters. So, the base of the image we see is approximately 10 meters across.

5 IDS 101 Sizes of things page 5 The next image shows a closer view of Juno. The distance across the field of view is about 1 meter.

6 IDS 101 Sizes of things page 6 The last image was very difficult to get because Juno liked to lick the camera. Here the field of view is 0.1 meters, or 10 centimeters. We could say that the first image of Juno is 10 1 meters across, which is the same thing as saying 10 meters across. The second image is 10 0 meters= 1 meter across. The third image is 10-1 meters = 0.1 meters. So, the distances across the image vary by the number of the exponent on the ten. See the table below for additional information on powers of 10: Exponent Number , , ,

7 IDS 101 Sizes of things page 7 Big deal concept: This is a convenient place to introduce the term factor of. We can say that 100 is a factor of ten greater than 10, or we can say that 20 is a factor of 2 larger than 10. We can also say that to increase 10 by a factor of 3 would be. This terminology will be used frequently so if you do not understand it, please contact an instructor. Next, go to a web site (the URL is below) and view a series of drawings and photographs that vary in distance across the image by powers of ten. The first image will be a drawing of what the Milky Way Galaxy may look like from a distance of 10 million light years from the Earth. The field of view is meters across (the whole galaxy looks very small). The Java program will load a series of drawings and photographs that are each 1/10 smaller (like cutting the paper in a piece that is only 10% of the whole piece). Notice how the values in the lower left change as the image converges onto a leaf on an oak tree in Florida. The process continues as the view goes to microscopic views inside the leaf. Part II For the last few pages, you have been looking at measurements made by other people. Now it is time to make some measurements of your own. You will need a marble (or a small ball). 1. Use the marble (and only the marble) to measure the horizontal width of this piece of paper. Again, you are not allowed to use any other tool besides the marble. What is the width of this piece of paper? What units are you forced to use? 2. How useful would this system of units be for conveying information to other people, perhaps in another country where they wouldn t know how big letter sized paper is? What would we have to do to make this system of units more useful? 3. Now use only your marble to measure the longer edge of this piece of paper. How long is the piece of paper?

8 IDS 101 Sizes of things page 8 4. Now measure the area of this surface of this piece of paper. Use only your marble. How would you do this? Do you have to make any more measurements? 5. What are the units of your answer to #4? What do those units mean? 6. Imagine somebody told you that the number of people in the next room was 40. Now imagine that somebody told you that the size of the next room was 40. The first statement probably sounds possible but the second one sounds wrong. Explain why it is impossible. Length: Circle the units below that are measurements of length. inch meters 2 liter mile kilometer cubic centimeter league yard Area is another quantity of great importance in all of science. Area is the size of a figure in a two dimensional space or plane. Imagine we want to determine the area of this classroom to order flooring materials. What process would we use? What is another technique for measuring the area of this classroom?

9 IDS 101 Sizes of things page 9 If you are going to buy carpet for your home you will need to know the floor area to be covered by the carpet. Let s say that you have a room that is 10 feet by 10 feet. To determine the area of the room we multiply two sides of the room together to get an area of 100 square feet. Notice that units also change! They are square feet, not just feet. Let s explore this a bit further. You have just remodeled your house so that the dimensions of the room are twice as long (20 feet X 20 feet). You order twice as much carpet for your remodeled room (assume you are discarding the old carpet). Is this the correct amount of carpet to order? Explain your answer. If the amount of carpet is not correct, what amount would be needed? If your room is 40 feet X 40 feet, how much carpet would you need? What is the mathematical relationship of the sides of the rectangle to the area of the rectangle that is, if we double the length of sides, how does the area of the rectangle change? In the Powers of 10 web site, if the image shown on the web site has a length of 10 0 meter (1 meter), what is the area of the field of view? Go to the next image that is 10 1 meters (10 meters) on a side. What is the area of this field of view? Go to the next image that is 10 2 meters (100 meters) on a side. What is the area of this field of view?

10 IDS 101 Sizes of things page 10 If we increase the length of the sides of a square by ten times, how much does the area change? Which of the following units are units of area? (circle the area units) meters 2 millimeter liter cubic centimeters acre square inches yard cm 2 Part III Volumes Volume is a measure of how much space something occupies. How would we measure the volume of this room? Describe a second method of measuring the volume of the room? (By the way, volume and mass are not the same thing. If an iron bar and a wooden board had exactly the same size, thickness, and shape then they would have the same volume. The bar would have more mass because iron is much more dense than wood. We'll learn more about mass later.) When we measure length, area, and volume we are measuring different things. We have already determined that the inch is a unit of length and square inches is a unit of area, so: Circle the units of volume listed below: Gallon yard cubic centimeters milliliters cm 2 cubic feet acre-foot km 3

11 IDS 101 Sizes of things page 11 Imagine an irregularly shaped solid rock. How could we determine the volume of the rock? (you don t have to measure the actual volume we want to know the method). LENGTH, AREA, and VOLUME (Part IV) We are now going to investigate the relationships between length, area, and volume for a cube. You will generate a lot of numbers, but the numbers themselves aren't important. You will calculate areas and volumes using mathematical formulas, but you do not need to memorize any formula. The point of this exercise is to look for patterns in the areas, volumes, and numbers that we calculate. In order to do that, we will compare the numbers and make graphs. Each group will divide into three subgroups: a) The perimeter group, b) The area group, c) The volume group. Each subgroup will complete one column of the Data Table below. Once all of the subgroups are finished, the subgroups will share their data. Once you have all of the data, create three graphs: one of each the perimeter, area, and volume of the cube as a function of the length of one side.

12 IDS 101 Sizes of things page 12 DATA TABLE: Geometry books usually describe the size of a cube by the length of a side, s. That description will work fine for us as well. The first thing you need to know is how to find the perimeter of one face, the surface area, and the volume of a cube from the length of one side. You can use the following mathematical formulas. (Don t worry about memorizing them! You don t need to know these. We are learning about length, area, and volume, not about cubes.) Length of a side = s Perimeter of one face = (4) s Total Surface Area = (6) s 2 Don t worry about where these formulas came from, but trust us. They are right. They may look funny since you may be used to seeing formulas for squares and not cubes, but these are correct. Volume = s 3 Use these formulas to fill in the table below (you only need to work on the column that you are responsible for, and you can copy the other results from your classmates.) Length of one side in cm Perimeter of one face in Surface area (total) in Volume in

13 IDS 101 Sizes of things page 13 GRAPHING: After you have made your graphs, get together with the other people in your group and answer these questions Questions for Review: Look at your figures and graph for the cube and answer the following: When the length of a side was increased from 1 cm to 2 cm, the length increased by a factor of 2: The perimeter increased by a factor of. The surface area increased by a factor of. The volume increased by a factor of. When the length of a side was increased from 3 cm to 6 cm, the length increased by a factor of 2: The perimeter increased by a factor of. The surface area increased by a factor of. The volume increased by a factor of. When the length of a side was increased from 2 cm to 6 cm, the length increased by a factor of 3: The perimeter increased by a factor of. The surface area increased by a factor of. The volume increased by a factor of. When the length of a side was increased from 2 cm to 8 cm, the length increased by a factor of 4: The perimeter increased by a factor of. The surface area increased by a factor of. The volume increased by a factor of. Answer the following questions in words (and symbols if you want). If we measure the length of a side of a cube in units of centimeters, then We measure the perimeter (or circumference) in units of. We measure the surface area in units of. We measure the volume in units of.

14 IDS 101 Sizes of things page 14 Would this same pattern of length, area, and volume hold for spheres? Let s find out by doing some measurements of metallic spheres (ball bearings). You will find a box with four different sizes of ball bearings. Select one of each size of ball bearing and return to your group. DATA TABLE: Measure the diameter of the ball bearings with the digital calipers on the lab cart. Enter the values below in ascending order. # Diameter (cm) Once you determine the diameter of each ball bearing, estimate how many of the smallest ball bearings would be required to fit in a sphere with the same volume as each of the larger ball bearings? (Use just the diameters to determine the answers). Estimates # of smallest ball bearings A good way to check your estimation of the number of the smallest ball bearings needed to equal the same volume as each of the other ball bearings is by immersing the ball bearings in a graduated cylinder. Pick a graduated cylinder and gently fill it with 50 milliliters of water. It is important to pour the water slowly into the cylinder so that you can minimize the formation of air bubbles in the water. Make sure that you gently tap the sides to get the air bubbles out of the water. If you drop them in the water, it could splash the water out and you will have to start over. 1) Now add ball bearing #2 by tipping the cylinder and gently rolling the ball bearing into the water. Carefully note the increase in volume of the water. Pour the water out and remove bearing #2 from the cylinder. 2) Fill the graduated cylinder to the 50ml mark again. This time add the smallest ball bearings one at a time (gently!) until you get the same increase in volume as when you added bearing #2.

15 IDS 101 Sizes of things page 15 Question: How many smallest ball bearings did you have to add to get the same increase in volume as when you added bearing #2? Repeat this with ball bearings 3 and 4 and record your data below. Results of Measurement Big Ball Bearing # Diameter of ball bearing # of smallest ball bearings that fit into big ball bearing. Do your answers match reasonably well with the estimates you made earlier? Geometry books often describe the size of a sphere by the radius, r. The radius only stretches halfway across the sphere. We will describe the size of a sphere by the diameter, d for this exercise. Diameter = d Circumference = (3.14) d Surface Area = (3.14) d 2 Volume = (0.523) d 3 Don t worry about where these formulas came from, but trust us. They are right. They may look funny since you may be used to seeing formulas for circles using the radius, but these are correct. Divide the task of determining the circumference, surface area, and volume of each of the ball bearings. Use these formulas to fill in the table below and you can copy the other results from your classmates.

16 IDS 101 Sizes of things page 16 Diameter in cm Circumf. in Surf. area in Volume in Look at your figures for the spheres and answer the following: When the diameter increased by a factor of 2: The circumference increased by a factor of. The surface area increased by a factor of. The volume increased by a factor of. When the diameter increased by a factor of 3: The circumference increased by a factor of. The surface area increased by a factor of. The volume increased by a factor of. Now take a few minutes to write out, in your own words, the pattern you have found examining the various lengths, areas, and volumes of the cubes and spheres. See an instructor before proceeding.

17 IDS 101 Sizes of things page 17 Part V- Application to Irregular Solid objects Do these same principles discovered in Part IV apply to irregularly-shaped solid objects? Let s see. On the lab cart you should find boxes of bolts like the ones illustrated below: Measure the length of each bolt and write the data below: A: B: C: D: Prediction: How many of the D bolts will be needed to equal the same volume as the following bolts: Explain your logic. C: B: A:

18 IDS 101 Sizes of things page 18 Check your predictions by comparing the volumes of the bolts. You may use a graduated cylinder and water (or maybe another method - ask your teacher). Report your results below: