/- / 18 Math A&l Name: '30 . Test on Exponential and Logarithmic Functions Points: [37 1. Let the function h(x) represent the height in centimetres of a cylindrical tin can with diameter x cm. hOC) =+0.5for4gxgl4_ a. Find the range of h. [3 marks] The function h1 is the inverse function of h. b. Find h_1(10). [2 marks] c. In the context of the question, interpret your answer to part (b). [1 mark] . d. Write down the range of h'l. [1 mark] IB Math A&I Name: lectures. 2. Professor Wei observed that students have difficulty remembering the information presented in his He modelled the percentage of information retained, R, by the function R(t) = 100empt, t 2 0, where t is the number of days after the lecture. He found that 1 day after a lecture, students had forgotten 50% of the information presented. a. Find the value of p. [2 marks] b. Use this model to find the percentage of information retained by his students 36 hours after Professor Wei's lecture. [2 marks] Based on his model, Professor Wei believes that his students will always retain some information from his lecture. c. State a mathematical reason why Professor Wei might believe this. [1 mark] d. Write down one possible limitation of the domain of the model. [1 mark] ... ... ... ... ... ... ........ ..... ...... .......... . ... ... . ... ........so ....... ...... ... .......... ................as .............. . ... ........ .................. . . .. ..... . ..... ..... ... . ................ . ........ ... . .. . .. ... ... ...IB Math A&I Name: 3. Jean-Pierre jumps out of an airplane that is flying at constant altitude. Before opening his parachute, he goes through a period of freefall. function. Jean-Pierre's vertical speed during the time of freefall, S, in m s-1, is modelled by the following S(t) = K - 60(1.2-t) , t20 where t, is the number of seconds after he jumps out of the airplane, and K is a constant. A sketch of Jean-Pierre's vertical speed against time is shown below. S(t) Jean-Pierre's initial vertical speed is 0 m s-1. a. Find the value of K. [2 marks] b. In the context of the model, state what the horizontal asymptote represents. [1 mark] c. Find Jean-Pierre's vertical speed after 10 seconds. Give your answer in km h-1 [3 marks] .. . ... ... .. ..... ......... .. .. ........ .................... .........mrs... . ........ .. ... ..... . ..... ...... .. . .. .......IB Math A&l Name : 4. Little Green island originally had no turtles. After 55 turtles were introduced to the island, their population is modelled by N (t) = a x 2-t+10, t20, where a is a constant and t is the time in years since the turtles were introduced. a. Find the value of a. [2 marks] b. Find the time, in years, for the population to decrease to 20 turtles. [2 marks] There is a number m beyond which the turtle population will not decrease. c. Find the value of m. Justify your answer. [2 marks] ... ... .... ... ............. .. ... . .............. ... .... ... . .. ............ .... ... ............. ...... ............... ..... ......... ... ... ....... ...... ... ..... ... .. .............. .. . . . .. ........\"a Math A&I Name: 3 5. ConSIder a functionx) , for 2 S x S 2 . The following diagram shows the graph off. J 'A x 3. . ,e .__ ., , 7.__._.- a. On the grid above, sketch the graph off'l. [4 marks] b. Write down the range off'l. [1 mark] [1 mark] c. Write down the value off'1(0). ......................................................................................................................................................................................... ......................................................................................................................................................................................... ......................................................................................................................................................................................... ......................................................................................................................................................................................... ......................................................................................................................................................................................... ......................................................................................................................................................................................... ......................................................................................................................................................................................... ......................................................................................................................................................................................... ......................................................................................................................................................................................... '5 Math A&| Name: 6. The followmg function models the growth of a bacteria population in an experiment P(t)=AX2', tZO where A IS a constant and t is the time, in hours, since the experiment began. Four hours after the experiment began, the bacteria population is 6400. 3. Find the value ofA. [2 marks] b. Interpret what A represents in this context. [1 mark] c. Find the time since the experiment began for the bacteria population to be equal to 40A. [3 marks]