Math 103 College Algebra and Trigonometry Exam 3 review problems Problem 1 The function f x x3 5 is one to one Find its inverse function and check your answer Problem 2 a Graph the function f x 3x b Use your graph from part a to graph the function g x 3 x c Use your graph from part b to graph the function h x 3 x 2 2 Problem 3 Solve 3x 27x 99 Problem 4 What is the domain of the function f x ln x Problem 5 What is the domain of the function f x log281 x2 4x 5 Hint You will need to solve a polynomial inequality Problem 6 Find the exact value of 3 log12 2 log12 18 without using a calculator You can use a calculator to check your answer of course Problem 7 Solve log2 x 1 log2 x 3 3 Check your answer Remember to make sure your solutions are in the domain of both log2 x 1 and log2 x 3 so that they really are solutions to the original equation You might have to discard solutions that aren t in the domain Problem 8 You invest 1000 at 5 compounded semiannually twice a year How long will it take until you have at least 1300 Problem 9 How much money should you invest now in order to get 500 after 5 years at 3 compounded continuously Problem 10 A dose of 13 mg of morphine is injected into a patient s bloodstream The amount of morphine in the patient s bloodstream follows an exponential decay model with a half life of 2 9 hours How much morphine is in the patient s bloodstream after 1 hour After how much time has the amount of morphine in the patient s bloodstream fallen to 10 of the initial dose Problem 11 A certain quantity of a radioactive substance is produced in a nuclear fission reaction The quantity of this substance decays over time following an exponential decay model It is found that there are 78 25 grams of this substance remaining 2 hours after the reaction and 4 hours after the reaction there are 72 03 grams left a Find the value of k in the exponential decay model for this substance Hint It is easiest to let t 0 correspond to the point in time 2 hours after the reaction so that you know the value of A0 If you do this remember that the point in time 4 hours after the reaction will not be t 4 b How much of this substance was originally produced in the reaction If in part a you used t 0 to represent the point in time 2 hours after the reaction then the first thing you must determine is what value of t corresponds to the reaction itself c What is the half life of this substance This substance in this problem is supposed to be zirconium 97 so you can check your answer by searching for the half life of zirconium 97 on the Internet The Wikipedia article Isotopes of zirconium is helpful Page 1 Problem 12 a Convert an angle of 225 to radians Express your answer as a multiple of in other words do not approximate your answer with a decimal b Convert an angle of 7 radians to degrees 12 15 Problem 13 Suppose is an acute angle and tan Find the values of the other 8 five trigonometric functions of Problem 14 Find the exact value of sec3 57 sin 33 tan2 57 without using a calculator Hint Use the fundamental identities and the complementary angle theorem Problem 15 Find the exact value of sin cos2 without using a calculator 6 4 Problem 16 There is a flat field with a single large oak tree in the middle Directly east of the tree some distance away is a white fencepost A surveyor is standing 360 yards directly south of the tree She measures the angle between her line of sight to the tree and her line of sight to the fencepost to be 42 1 To the nearest yard how far is the fencepost from the tree How far is the surveyor from the fencepost Here is some graph paper for you to use for something if you like y y 5 5 4 4 3 3 2 2 1 5 4 3 2 1 1 1 2 3 4 5 x 5 4 3 2 1 1 1 1 2 2 3 3 4 4 5 5 Page 2 2 3 4 5 x