A prism that has circles as bases is called a cylinder. Recall that the formula for any prism is V = A b h. Since the area of the circular base is A = πr 2 , we can replace πr 2 for A b in the formula, giving us V = πr 2 h,where r is the radius of the circular base, and h is the height of the cylinder. A sphere is a three-dimensional object that has no sides. A basketball is a good example of a sphere. The vol- ume of a sphere is given by the formula V = ᎏ 4 3 ᎏ πr 3 . Example Determine the volume of a sphere whose radius is 1.5'. Replace 1.5' in for r in the formula V = ᎏ 4 3 ᎏ πr 3 . V = ᎏ 4 3 ᎏ πr 3 V = ᎏ 4 3 ᎏ π(1.5) 3 V = ᎏ 4 3 ᎏ (3.375)π V = 4.5π≈14.14 The answer is approximately 14.14 cubic feet. Example An aluminum can is 6" tall and has a base with a radius of 2". Determine the volume the can holds. Aluminum cans are cylindrical in shape, so replace 2" for r and 6" for h in the formula V = πr 2 h. V = πr 2 h V = π(2) 2 (6) V = 24π≈75.40 cubic feet Cylinder V = πr 2 h r h – THEA MATH REVIEW– 170 Data Analysis Data analysis simply means reading graphs, tables, and other graphical forms. You should be able to: ■ read and understand scatter plots, graphs, tables, charts, figures, etc. ■ interpret scatter plots, graphs, tables, charts, figures, etc. ■ compare and interpret information presented in scatter plots, graphs, tables, charts, figures, etc. ■ draw conclusions about the information provided ■ make predictions about the data It is important to read tables, charts, and graphs very carefully. Read all of the information presented, pay- ing special attention to headings and units of measure. This section will cover tables and graphs. The most com- mon types of graphs are scatter plots, bar graphs, and pie graphs. What follows is an explanation of each, with examples for practice. Tables All tables are composed of rows (horizontal) and columns (vertical). Entries in a single row of a table usually have something in common, and so do entries in a single column. Look at the table below that shows how many cars, both new and used, were sold during particular months. Month New Cars Used Cars June 125 65 July 155 80 August 190 100 September 220 115 October 265 140 Tables are very concise ways to convey important information without wasting time and space. Just imag- ine how many lines of text would be needed to convey the same information. With the table, however, it is easy to refer to a given month and quickly know how many total cars were sold. It would also be easy to compare month to month. In fact, practice by comparing the total sales of July with October. In order to do this, first find out how many cars were sold in each month. There were 235 cars sold in July (155 + 80 = 235) and 405 cars sold in October (265 + 140 = 405). With a little bit of quick arithmetic it can quickly be determined that 170 more cars were sold during October (405 – 235 = 170). – THEA MATH REVIEW– 171 Scatter Plots Whenever a variable depends continuously on another variable, this dependence can be visually represented in a scatter plot. A scatter plot consists of the horizontal (x) axis, the vertical (y) axis, and collected data points for vari- able y, measured at variable x. The variable points are often connected with a line or a curve. A graph often con- tains a legend, especially if there is more than one data set or more than one variable. A legend is a key for interpreting the graph. Much like a legend on a map lists the symbols used to label an interstate highway, a rail- road line, or a city, a legend for a graph lists the symbols used to label a particular data set. Look at the sample graph below. The essential elements of the graph—the x- and y-axis—are labeled. The legend to the right of the graph shows that diamonds are used to represent the variable points in data set 1, while squares are used to rep- resent the variable points in data set 2. If only one data set exists, the use of a legend is not essential. (Note: This data was used in the above example for tables.) The x-axis represents the months after new management and promotions were introduced at an automo- bile dealership. The y-axis represents the number of cars sold in the particular month after the changes were made. The diamonds reflect the New Cars sold and the squares show the number of Used Cars sold. What conclusions can be drawn about the sales? Note that the New and Used car sales are both increasing each month at a pretty steady rate. The graph also shows that New Cars increase at a higher rate and that there are many more New Cars sold per month. Try to look for scatter plots with different trends—including: ■ increase ■ decrease ■ rapid increase, followed by leveling off ■ slow increase, followed by rapid increase ■ rise to a maximum, followed by a decrease ■ rapid decrease, followed by leveling off ■ slow decrease, followed by rapid decrease ■ decrease to a minimum, followed by a rise ■ predictable fluctuation (periodic change) ■ random fluctuation (irregular change) 0 50 100 150 200 250 300 12345 Months New Cars Used Cars Cars Sold – THEA MATH REVIEW– 172 Bar Graphs Whereas scatterplots are used to show change, bar graphs are often used to indicate an amount or level of occur- rence of a phenomenon for different categories. Consider the following bar graph. It illustrates the number of employees who were absent due to illness during a particular week in two different age groups. In this bar graph, the categories are the days of the week, and the frequency represents the number of employ- ees who are sick. It can be immediately seen that younger employees are sick before and after the weekend. There is also an inconsistent trend for the younger employees with data ranging all over the place. During mid-week the older crowd tends to stay home more often. How many people on average are sick in the 41–65 age group? To find the average you first must find out how many illnesses occur each week in the particular age group. There are a total of 41 illnesses for a five-day period (3 + 10 + 12 + 12 + 4 = 41). To calculate the average, just divide the total illnesses by the number of days for a total of 8.2 illnesses ( ᎏ 4 5 1 ᎏ = 8.2) or more realistically 8 absences per day. Pictographs Pictographs are very similar to bar graphs, but instead of bars indicating frequency, small icons are assigned a key value indicating frequency. In this pictograph, the key indicates that every icon represents 10 people, so it is easy to determine that there were 12 ϫ 10 = 120 freshman, 5.5 ϫ 10 = 55 sophomores, 5 ϫ 10 = 50 juniors, and 3 ϫ 10 = 30 seniors. Freshmen Sophmores Juniors Seniors Number of Students at the Pep Rally Key: indicates 10 people 7 33 6 9 3 10 12 12 4 0 5 10 15 Monday Tuesday Wednesday Thursday Friday Number of Employees Who Are Sick 18–40 41–65 – THEA MATH REVIEW– 173 Pie Charts and Circle Graphs Pie graphs are often used to show what percent of a total is taken up by different components of that whole. This type of graph is representative of a whole and is usually divided into percentages. Each section of the chart rep- resents a portion of the whole, and all of these sections added together will equal 100% of the whole. The chart below shows the three styles of model homes in a new development and what percentage of each there is. Find the percentage of Estate homes. In order to find this percentage, look at the pie chart. The categories add up to 100% (25 + 30 + 45 = 100). From the actual chart you can visually see that 45% of the homes are done in the Estate model. Broken Line Graphs Broken-line graphs illustrate a measurable change over time. If a line is slanted up, it represents an increase, whereas a line sloping down represents a decrease. A flat line indicates no change. In the broken line graph below, the number of delinquent payments is charted for the first quarter of the year. Each week the number of outstanding bills is summed and recorded. There is an increase in delinquency for the first two weeks and then it maintains for an additional two weeks. There is a steep decrease after week 5 (initially) until the ninth week, where it levels off again but this time at 0. The 11th week shows a radical increase followed by a little jump up at week 12, and then a decrease to week 13. It is also interesting to see that the first and last weeks have identical values. Now, take the skills you have learned or honed in this review and apply them to the next practice test. 0 1 2 3 4 5 6 7 8 9 10 11 12345678910111213 Week Number Number of Customers Chateau 25% Estates 45% Models of Homes Provincial 30% – THEA MATH REVIEW– 174 U nlike math, writing is flexible. There are many different ways to convey the same meaning. The THEA Writing section tests your writing skills in two ways. First, it asks you approximately 50 mul- tiple-choice questions related to writing. You do not need to write out any sentences or paragraphs for these questions; you simply need to answer the questions correctly by choosing the answer a, b, c,or d. Each question is based on a passage (you are already familiar with the test format of a passage followed by questions from the Reading section, no doubt). However, the passages in the multiple-choice Writing section are shorter than most of those in the Reading section. Another difference is that each part of a passage in the Writ- ing section is assigned a number, so you can identify specific sentences or sentence fragments. The second part of the Writing section asks you to write an essay. This essay is evaluated based on your abil- ity to communicate effectively in writing. You will need to express yourself clearly and correctly in an essay of approximately 300–600 words. Keep in mind that an average page of handwritten material is approximately 225 words. More important than the essay’s length are its content and organization. As you study, you may be tempted to focus more on the multiple-choice questions than on the essay, sim- ply because it’s easier to tell whether you’re correct on a multiple-choice question. This would be a mistake. Your CHAPTER THEA Writing Review CHAPTER SUMMARY This chapter covers the topics that will help you succeed on the mul- tiple-choice and essay portions of the Writing test. You will learn about grammar, organization, as well as how to recognize your audience. 6 175 . bit of quick arithmetic it can quickly be determined that 170 more cars were sold during October (405 – 235 = 170 ). – THEA MATH REVIEW– 171 Scatter Plots Whenever a variable depends continuously. practice test. 0 1 2 3 4 5 6 7 8 9 10 11 12345 678 910111213 Week Number Number of Customers Chateau 25% Estates 45% Models of Homes Provincial 30% – THEA MATH REVIEW– 174 U nlike math, writing is. approximately 50 mul- tiple-choice questions related to writing. You do not need to write out any sentences or paragraphs for these questions; you simply need to answer the questions correctly by choosing