Precalculus Project

Name _________________________________ I.D. Number _______________________ Project 3 Evaluation 33 Precalculus: Analytic Geometry & Algebra (MTHH 043 060) Be sure to include ALL pages of this project (including the directions and the assignment) when you send the project to your teacher for grading. Don’t forget to put your name and I.D. number at the top of this page! This project will count for 10% of your overall grade for this course. Be sure to read all the instructions and assemble all the necessary materials before you begin. You will need to print this document and complete it on paper. Feel free to attach extra pages if you need them. To earn full credit, you must justify your solutions by showing your work. When you have completed this project, you may submit it electronically through the online course management system by scanning the pages into either .pdf (Portable Document Format), or .doc (Microsoft Word document) format. If you scan your project as images, embed them in a Word document in .gif image format. Using .gif images that are smaller than 8 x 10 inches, or 600 x 800 pixels, will help ensure that the project is small enough to upload. Remember that a file that is larger than 5,000 K will NOT go through the online system. Make sure your pages are legible before you upload them. Check the instructions in the UNHS online course management system or the enrollment information provided with your print materials for more information. Remember that showing your work in arriving at solutions is required for all problems. All graphs must be done on the grids provided or on graph paper. Computer-generated graphs will not be acceptable. Methods learned in this course should be used when solving the problems on this project. Part A: Procedural Skill & Fluency 1. Sketch the graph of f(x) = 3− x . Indicate the domain and range. Project 3 MTHH 043 2. Sketch the graph of g(x) = log3 (x − 1) . Indicate the domain and range. 1 as an equivalent logarithmic equation. 9 3. Rewrite 3−2 = 4. Rewrite log5 5. Find the solution set of log6 (x + 1) + log6 x = 1 . 1 = −2 as an equivalent exponential equation. 25 Use either the substitution method or the elimination method to find the solution set of each system of equations. Sketch the graphs of the systems in problems 6, 7, and 8. 6. Project 3 2x + y = 10 x – 2y = -5 MTHH 043 7. 8. y = x2 y−9 =0 x2 + y2 = 16 y + 4 = x2 Sketch the region defined by each system of inequalities. 9. Project 3 3x − y  1 x + 2y  5 MTHH 043 10. 11. x2 +y 2  16 y  x For the following system of equations, find the augmented matrix, reduce it to an echelon matrix, and find the solution set. x − 2y + z = 7 3x − 5y + z = 14 2x − 2y − z = 3 Find the solution set of each equation. 12. 25x = 5x+2 13. log4 (x − 3) = − 1 2 Given: logb 2 = 0.4, logb 3 = 0.5, and logb 5 = 0.6. Use properties of logarithms to find the following values. 14. logb 60 15. logb 0.6 Project 3 MTHH 043 Part B: Applications 1. The fox population in a certain region has an annual growth rate of 9% per year. In the year 2012, there were 23,900 fox counted in the area. What is the fox population predicted to be in the year 2020? 2. A car was valued at $38,000 in the year 2007. By 2013, the value had depreciated to $11,000 If the car’s value continues to drop by the same percentage, what will it be worth by 2017? 3. Alyssa opened a retirement account with 7.25% APR in the year 2000. Her initial deposit was $13,500. How much will the account be worth in 2025 if interest compounds monthly? How much more would she make if interest compounded continuously? 4. Drew wants to save $2,500 to go to the next World Cup. To the nearest dollar, how much will he need to invest in an account now with 6.25% APR, compounding daily, in order to reach his goal in 4 years? 5. Technetium-99m is used as a radioactive tracer for certain medical tests. It has a half-life of 1 day. Consider the function 𝑇 where 𝑇(𝑑) = 100(2)−𝑑 is the percent of Technetium-99m remaining 𝑑 days after the test. How many days until only 5% remains? 6. Radioactive Iodine has a half-life of 13 hours. Consider the function 𝐼(ℎ) = 100 ∙ (2)−ℎ/13 is the mg of Iodine remaining ℎ hours after the initial measure. How many hours until only 10 mg remains? Show all your work. 7. In a lake, the population of a species of fish is 400 and is growing by 2% each day. According to the model, when will the population reach 1500? Project 3 MTHH 043 8. A patient takes 400 mg of medicine. The amount in her bloodstream is decreasing by 8% every hour. According to the function, how long until 100 mg is remaining? 9. You are offered two jobs selling dental supplies. One company offers a straight commission of 6% of sales. The other company offers a salary of $350 per week plus 3% of sales. Write and solve a system of equations to find how much you would have to sell in a week to make the straight commission offer better. 10. The perimeter of a rectangle is 30 meters and the length is 3 meters greater than the width. Write and solve a system of equations to find the dimensions of the rectangle. 11. A rectangular tract of land has a perimeter of 40 kilometers and an area of 96 square kilometers. Write and solve a systems of equations to find the dimensions of the land. 12. An airplane flying into a headwind travels the 1800-mile flying distance between Pittsburgh, Pennsylvania and Phoenix, Arizona in 3 hours and 36 minutes. On the return flight, the distance is traveled in 3 hours. Write and solve a system of equations to find the airspeed of the plane and the speed of the wind. 13. At a local high school city championship basketball game, 1435 tickets were sold. A student admission ticket cost $1.50 and an adult admission ticket cost $5.00. The total ticket receipts for the basketball game were $3552.50. Write and solve a system of equations to find how many of each type of ticket were sold. 14. An inheritance of $12,000 was invested among three funds: a money-market fund that paid 5% annually, municipal bonds that paid 6% annually, and mutual funds that paid 12% annually. The amount invested in mutual funds was $4,000 more than the amount invested in municipal bonds. The total interest earned during the first year was $1,120. Write and solve a system of equations to find how much was invested in each type of fund. Project 3 MTHH 043 15. The salary for each of the stars of a new movie is $5 million, and the supporting actors each receive $1 million. The total amount spent for the salaries of the actors and actresses is $19 million. If the cast has 7 members, use Cramer’s Rule to find the number of stars and supporting roles in the movie. 16. A vendor sells small drinks for $1.15, medium drinks for $1.75, and large drinks for $2.25. During a week in which he sold twice as many small drinks as medium drinks, his total sales were $2,238.75 for 1385 drinks. Use Cramer’s Rule to determine how many of each drink were sold. Part C: Matrices and Cryptography Cryptography is the study of coded messages. Matrices can be used to code messages so that they can only be read after being deciphered by a key. The first step is to decide on a key that can be used to encode the matrix. The key must be an n x n invertible matrix. The next step is to convert the message to numbers and write it as a matrix. Each letter of the alphabet is represented by a number (A is 1, B is 2, …). The number 0 is used to represent a blank space. Finally, the message is encoded by multiplying it by the key. Activity 1 1 −2 Use [ ] to encode the message “SATURDAY AT −1 3 NOON” Step 1: Convert the message to numbers and write it as a matrix. Remember to use the number 0 for a space between each word. The first 3 were done for you. S-19 A-1 T-20 The key is a 2×2 matrix. To make the matrix multiplication possible, write the message as an 8×2 matrix. 𝑆 𝑇 𝑅 𝐴 𝑇 𝑁 𝑂 19 20 𝐴 𝑈 𝐷 𝑌 𝐴 𝑂 𝑁 Project 3 [ 1 ] MTHH 043 Step 2: Multiply the converted message by the key. 19 20 1 18 −35 −1 1 −2 × [ ]= −1 3 [ ] [ ] Step 3: Remove the matrix notation to reveal the encoded message. 18 -35 -1 Activity 2 3 5 Use [ ] to encode each message. −2 −3 1. CALL ME 2. SEE YOU LATER 3. ORDER PIZZA Activity 3 To decode a message, the inverse of the key must be found. The coded message is then written in matrix form to make the multiplication possible. For instance, if the key is an 𝑛 × 𝑛 matrix, the message is written as 𝑘 × 𝑛 matrix, where 𝑘 is the number of rows necessary to include each number in the matrix. If there are not enough characters to fill a row, insert “0’s” as spaces. Finally, the coded matrix is multiplied by the inverse of the key. How to find the Inverse of a 2×2 matrix: Project 3 MTHH 043 3 2 If [ ] was used to encode the message 62 63 46 48 13 65 59 100 40 135, Use the inverse to 1 5 decode the message. 5/13 −2/13 Step 1: Find the Inverse using the steps above [ ] −1/13 3/13 Step 2: Arrange the encoded matrix into a 2 × 𝑛 matrix 62 63 46 48 13 65 [ ] Step 3: Multiply the converted message by the inverse 19 62 63 14 46 48 5/13 −2/13 13 65 × [ ]= −1/13 3/13 [ ] [ 5 4 ] Step 4: Remove the matrix notation to reveal the message. 19 5 14 4 Step 5: Change the numbers back to letters. SEN Project 3 MTHH 043 Activity 4 12 −7 Use the inverse of [ ] to decode each message. −5 3 1) 151 −88 150 −86 93 −54 −35 22 −5 3 191 −111 −30 18 182 −105 2) −27 17 38 −21 84 −49 21 −11 131 −76 201 −116 161 −93 1 2 Use the inverse of [ ] to decode the message. 3 5 3) 11 21 64 112 25 50 29 53 23 46 40 75 55 92 Part D: Logarithm Investigations 1. Use your calculator to complete the table. Record the values to three decimal places. 2. Look closely at the values for the logarithms in the table. Look for pairs of values that add up to a third value in the table. For example, add log 2 and log 3. Where can you find that sum in the table? Record the equations that you find in the form log 2 + log 3 = log ?. (Hint: You should find at least six equations.) 3. Write a conjecture based on your results from part 2. Project 3 MTHH 043 4. Use your conjecture to write log 90 as the sum of two logs. Do the same for log 30 and log 72. Then use the table and your calculator to test your conjecture. Log 90 = log __ + log ___ Log 30 = log __ + log ___ Log 72 = log __ + log ___ 5. Now find pairs of values in the table that subtract to equal another value in the table. Record your results in the form log 9 – log 3 = log ?. Describe any patterns you see. 6. Write a conjecture based on your results from part 5. 7. Now find values in the table that can be multiplied by a small integer to give another value in the table, such as 3 log 2 = log ?. Describe any patterns you see. You may want to think about different ways to express numbers such as 25 or 27 using exponents. 8. Use your conjectures to find each of the following: a. log 3 + log 7 = log ? b. log 80 – log 20 = log ? c. 3 log 4 = log ? d. log 4 + log ? = log 44 e. log 95 – log ? = log 19 f. 2 log ? = log 400 Project 3 MTHH 043 Declaration I know that plagiarism is wrong. Plagiarism is to use another’s work and pretend that it is one’s own. I declare that this is my own work. Signature _________________________________ Student I.D. number ________________ Project Submission This project can be submitted electronically. Access the online version of your course and follow the directions provided for Project 1. (Go to Project 3 in Unit 5 on the navigation panel.) For mail submission directions, refer to the enrollment information included with your print materials. Project 3 MTHH 043

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