NAME:Math 103 Exam 3 retake12 December 2008100 pointsInstructions:1. This exam has 7 pages (including this one), which contain 8 problems and one bonus problem.Please check that you have all of the pages.2. Answer all of the following questions clearly and completely. Justify all of your answers.3. You may not use a book or any notes for this exam.4. Give your answer to each problem completely and clearly in the space provided. You may usethe back of the exam pages for scratch work; however, if you want this work to be considered,make note of it in the space provided for the problem.5. Erase or cross out work you do not wish to be graded.6. Credit, partial or full, will be given only if sufficient steps leading to the answers are shown.7. You have 50 minutes to complete this exam.Page 1Problem 1. (10 points) Be sure to show work for both parts of this problem, even if you can dothem in your head, so that I can see that you understand how to do these conversions and that youaren’t just using your calculator.(a) (5 points) Convert4π3radians to degrees.(b) (5 points) Convert 165◦to radians. Express your answer as a multiple of π (in other words, donot approximate your answer with a decimal).Problem 2. (12 points) Find the exact value of 2 log345 − log325 without using a calculator.Page 2Problem 3. (12 points) Suppose θ is an acute angle and cos θ =2853. Find the values of the otherfive trigonometric functions of θ.Problem 4. (14 points) The function f (x) =3x − 10is one-to-one. Find its inverse function f−1and check your answer.Page 3Problem 5. (14 points) A horizontal awning 10 feet wide is to be attached to the side of a building.A supporting chain will run from the outer edge of the awning to the side of the building above thepoint the awning is attached. The angle between the awning and the chain must be at least 37◦inorder to support the weight of the awning. The builder wants to use the least possible amount ofchain.(a) (7 points) How long should the chain be?(b) (7 points) How far above the awning should the chain be attached to the building?Page 4Problem 6. (16 points) Remember that radioactive decay follows an exponential decay model, sothe amount A(t) of a radioactive substance at time t is given byA(t) = A0ekt,where A0is the amount of the substance at time t = 0. A sample of 100 grams of radioactiverhenium-182 is produced in a reaction. After 22 hours, there are 78.80 grams of rhenium-182remaining.(a) (8 points) Find the value of k in the exponential decay model above.(b) (8 points) What is the half-life of rhenium-182?Page 5Problem 7. (12 points)(a) (6 points) What is the domain of f(x) = ln x?(b) (6 points) The graph of a function g(x) is shown below. Based on this graph, what is thedomain of the function h(x) = lng(x)? Why?−11−11−22−22−33−33−44−44−55−55xyy = g(x)Page 6Problem 8. (10 points) Solvelog(3x + 157) + log(100) = 5.Bonus problem (+4 points). There are a few trigonometric functions which were once commonbut are now very rarely used. One of these is called the versine, written versin θ, which is defined asversin θ = 1 − cos θ.Find the exact value of (1 + sin 85◦)(versin 5◦)(csc25◦) without using a calculator.Page