Geoetry Unit Activity Answer Sheet Unit: Extending to Three Diensions This Unit Activity will help you eet these educational goals: Matheatical Practices You will use atheatics to odel real-world situations. Inquiry You will conduct online research. STEM You will apply atheatical and technology tools and knowledge to grow in your understanding of atheatics as a creative huan activity. Directions and Analysis Task : Finding Density All objects have ass. Mass is the aount of aterial in an object. Earlier in this unit, you studied the volue of objects. When you cobine these two concepts, you get a useful easureent called density. The density of an object is its ass per unit of volue. Iagine two objects in which the larger one actually weighs less than the saller one. This difference in weight is attributed to density. Consider a candle that has a volue of 96 cubic inches and a weight of 44.6 ounces. What is the density of the candle? The density is defined as its ass per unit volue, so siply divide the ass of the candle by its volue: ass 44.6 ounces density = 0.46 ounces/cubic inch. volue 96 cubic inches In this case, you are finding the average density of the candle. In ost cases, you will be finding the average density of objects because ost objects do not have a unifor density. For instance, the portion of the candle where the wick is ight have a different density than the rest of the candle. Now that you know about density, let s work on soe applications. Harold is a woodworker and is interested in finding the density of two different types of wood. He bought a pine log that has a 5-inch radius and is 30 inches long. The weight of the log is approxiately 42.63 pounds. He also has an oak board that is 5.5 inches wide,.5 inches thick, and 3 feet long. The board weighs approxiately 8.05 pounds. a. Which shapes can you use to odel the log and the board? I can odel the pine log using a right cylinder (radius = 5 inches and height = 30 inches). The odel for the oak board can be a rectangular pris (width = 5.5 inches, height =.5 inches, and length = 3 feet = 36 inches). 203 EDMENTUM, INC.
b. What is the density of the pine log? Show your work. I already know the weight of the pine log, so to find the density I need to find the volue. Because the log can be odeled by a cylinder: volue of log = volue of cylinder = r 2 h = (5 inches) 2 30 inches 2,356.2 cubic inches. Substituting into the density forula: ass 42.63 pounds density = volue 2,356.2 cubic inches 0.08 pounds/cubic inch. c. What is the density of the oak board? Show your work. I already know the weight of the oak board, so to find the density I need to find the volue. Because the board can be odeled by a rectangular pris: volue of the board = volue of a rectangular pris = = 36 inches 5.5 inches.5 inches = 297 cubic inches. l w h Substituting into the density forula: ass density = volue 8.05 pounds 0.027 pounds/cubic inch. 297cubicinches d. If you had a block of each type of wood with the sae diensions, how would their weights copare? Explain. If the blocks of wood had the sae diensions, they would have equal volues. However, because the density of oak is greater than the density of pine, the oak block would weigh ore than the pine block. Based on the densities found, the weight of the oak block would be.5 ties the weight of the pine block. e. Water has a density of about 0.036 pounds/cubic inch. You ight know that wood floats on water. What can you conclude about the densities of objects that float and the densities of objects that do not? Review your results fro parts b and c before deciding. The densities of both types of wood are less than the density of water. Any object whose density is less than the density of water will float in water, and any object whose density is greater than the density of water will sink in water. Task 2: The Story of Archiedes About 2,200 years ago, one of the ost faous atheaticians in history ade a ajor discovery. The atheatician was Archiedes, who was presented with a proble by King Hiero II of Syracuse in Sicily. 2
The king had given a bar of gold to a jeweler and ordered hi to ake it into a crown. After receiving the crown, the king suspected that the jeweler had replaced soe of the gold with an equal weight of a cheaper etal like silver and kept the reainder of the gold for hiself. The king had no way to deterine whether this was true, so he gave the crown to Archiedes and asked hi to devise a way to find out. At the tie, Archiedes knew that gold was ore dense than silver. So, if the volue of the crown was greater than the volue of a bar of gold, they would have proof that the jeweler had stolen soe of the gold. However, Archiedes had no way to find the volue of an irregular shape like a crown. Archiedes struggled with this proble for a long tie. One day, he went to take a bath. As he lowered hiself into the bath, he noticed that the water level rose. As he continued to lower hiself into the water, the water level continued to rise. He was so excited at his discovery that he juped out of the bath and, before he could reeber to put his pants on, went running through the streets yelling Eureka! Eureka! which in Greek eans I ve found it! I ve found it! He now had a way to easure the volue of the irregularly shaped crown. In Archiedes s case, the jeweler had indeed stolen gold fro the king. This did not end well for the jeweler. a. A key coponent to this story is that the jeweler deceptively replaced an equal weight (or equal ass) of gold with silver. How does this action result in increasing the volue of the crown? Explain using one or ore equations. Volue (V), density (ρ), and ass () are related through the equation Rearranging the equation gives V.. V The jeweler held constant by substituting an equal weight of silver, but did not consider how the density difference of the etals would change the volue. If density decreases, as in the case of replacing gold with silver, the volue ust increase. The crown would grow in volue in this way based on the jeweler s deceptive actions. b. What did Archiedes discover in the bath? Why was he so excited? Describe the discovery using geoetric ters. What do you predict Archiedes did next with his new discoveries? Explain. Archiedes discovered that the volue of water displaced by an object, in this case his body, is equal to the volue of the object. He was excited because he now had a way to easure the volue of an irregularly shaped object. Archiedes used water displaceent to find the volue of the crown. Subsequently, he was able to prove the crown was not ade purely of gold through the relationship. V 3
Task 3: Displaceent and Buoyancy Buoyancy is the ability to float in a liquid such as water. Archiedes s experience with water displaceent in the bath gave rise to Archiedes s Principle, which states that the upward buoyant force, or push, exerted on a body iersed in a fluid is equal to the weight of the fluid the body displaces. In technical ters, buoyancy is the upward force exerted by a fluid on a body when the body is iersed in the fluid. a. Making use of Archiedes s Principle, displaceent, and buoyancy, explain why soe objects float on water, but others do not. Use the diagra to help you decide. If the density of an object is less than that of water, the weight of water that the object displaces (if fully iersed) is greater than the weight of the object. So, if such an object is fully iersed in water, the upward buoyant force on it is greater than the downward force of its weight. The overall effect is an upward push on the object. The object rises and finally floats with a part of it iersed. If the density of an object is greater than that of water, the weight of water that the object displaces is less than the weight of the object. So, the buoyant force on the object is not enough to balance its weight. The object sinks under the net downward force on it. b. The Titanic is one of the ost faous passenger ships in history. Search the Internet to deterine how uch the Titanic weighed in etric tons. The Titanic weighed approxiately 46,000 etric tons when it was not loaded. c. The hull of a ship should be the only portion of a ship that is below the water. Therefore, at a iniu, how any tons of sea water should the hull of the Titanic displace while it s in water? At a iniu, the hull would need to displace 46,000 etric tons of sea water. 4
d. Search the Internet to find the density of surface sea water in kilogras/cubic eter. Answers will vary depending on the source. The density of sea water at the surface is approxiately,027 kilogras/cubic eter. e. Convert the density of surface sea water to etric tons/cubic eter. Use the Internet to look for the proper conversion factor, if needed. kg etric tons etric tons,027 0.00.027 3 3 kg The density of surface sea water is about.027 etric tons/cubic eter. f. Based on your calculations so far, how any cubic eters of water did the hull of the Titanic have to displace to stay afloat? The volue of 46,000 etric tons of sea water is given by: ass volue = density 46,000 etric tons.027 etric tons / cubic eter 44,790 cubic eters. Task 4: Density and Population The concentration of people or other species in a given area is called the population density. When dealing with population density, we use area instead of volue because it is a better easure of living space. For instance, the population density in a given city would be found by dividing the population of the city by the nuber of square iles that the city takes up. a. Use the Internet to find the population of Ohio. Then use this site to estiate the area of Ohio in square iles. What is your estiate? Use this area to estiate the population density of Ohio. According to the latest census, the population of Ohio is about,544,95 people. Estiates of the area will vary, but one possible estiate is 40,440 square iles. For this estiate, the population density is calculated as: population population density = area,544,95people 40,440 square iles 285.5 people per square ile. 5
b. Use the Internet to find the actual area of Ohio. Then find the actual population density of Ohio. How does this copare with your estiate? Why was your initial estiate greater or less than the actual population density? The actual area of Ohio is 44,820 square iles. Therefore, the actual population density is: population population density = area,544,95people 44,820 square iles 257.6 people per square ile. The initial estiate was greater than the actual population density because the area used in the estiation was saller. c. How do you think the population density of Cleveland, a ajor city in Ohio, copares with the population density of the state? Explain your answer. I would expect the population density in a ajor city to be greater than the population density of the whole state. In the rural parts of state, there will be places where people are uch ore spread out, whereas in the city there are lots of people confined to the liited area of the city. Resources Docuent any references you used for this project below. At iniu, include a title and URL for any Internet resource: 6
Evaluation This project will be evaluated on a rubric that is based on the copleteness, clarity, and thinking you exhibit in the Directions and Analysis section above. Total Points: 0 Task : Finding Density a. Identify geoetric odels for a log and a board. b. Find the density of the log. c. Find the density of the board. d. Copare two types of wood of equal volue. e. Conclude why wood floats on water. Task points: 3.5 Task 2: The Story of Archiedes a. Deterine how the ipurity of the crown increased its volue. b. Explain Archiedes discovery in the bath. Task points:.5 Task 3: Displaceent and Buoyancy a. Use displaceent and buoyancy to explain why objects float. b c. Deterine the weight of the Titanic and how uch water it displaced. d e. Convert the density of sea water to different units. f. Figure the volue of sea water the Titanic ight have displaced to stay afloat. Task points: 3 Task 4: Density and Population a. Calculate the population density of Ohio. b. Use actual area to find the population density of Ohio. c. Copare the population density of Cleveland with that of Ohio. Task points: 2 7