W6 - Connections with Exponential Models Unit 3 MCV4U Jensen 1) A 100-mg sample of thorium-233 (Th-233) is placed into a nuclear reactor. After 10 min, the sample has decayed to 73 mg. Use the equation N(t) = Noeat to answer the following questions: a) Determine the disintegration constant ) for Th-233. b) Determine the half-life of Th-233 c) Write the equation that gives the amount of Th-233 remaining as a function of time, in terms of half-life. d) How fast is the sample decaying after 5 min?2) Radon-222 (Rn-222) is a radioactive element that spontaneously decays into polonium-218 (Po-218) with a half-life of 3.8 days. The atoms of these two substances have approximately the same mass. Suppose that the initial sample of radon has a mass of 100 mg. I The mass of rad0n, in milligrams, as a function oftime is given by the function MR\" (t) = M0 6)\3) Consider a car shock absorber modelled by the equation Mt) = e'o'\" sin t, where hft) represents the vertical displacement, in meters, as a function of time, t, in seconds. Determine when the maximum vertical velocity, in m/s, occurs and its value, given that '90.\") = h'(t). 4) Rocco and Biff are two koala bears that are foraging for food together in a eucalyptus tree. Suddenly, a gust of wind causes Rocco to lose his grip and begin to fall. He quickly grabs a nearby vine and begins to swing away from the tree. Rocco's horizontal displacement as a function of time is given bv the . t _ . , . . equation x(t) = 5 cos (it?) e \"'1' where x IS Rocco 5 horizontal displacement from the bottom of his swing arc, in meters, at time t, in seconds. a) Biff can grab Rocco if Rocco swings back to within 1 meter from where he started falling. Will Biff be able to rescue Rocco? Explain, using mathematical reasoning. b) The other option Rocco has is to let go of the vine at the bottom of one of the swing arcs and drop to the ground. But Rocco will only feel safe doing this if his horizontal velocity at the bottom ofthe swing is less than 2 m/s. Assuming that Biff is unable to save his friend, how many times must Rocco swing back and forth on the vine before he can safely drop to the ground? S)The voltage signal from a standard North American wall socket can be described by the equation V(t) = 170 sin(1201tt), where t is time, in seconds, and V is the voltage, in volts, at time t. 3) Find the max and min voltage levels and the times at which they' occur. b) For the given signal, determine i) the period, T, in seconds ii) the frequency, f, in hertz iii] the amplitude, A, in volts 6) Consider a simple pendulum that has a length of 50 cm and a max horizontal displacement of 8 cm. a) Find the period of the pendulum. b) Determine a function that gives the horizontal position of the bob as a function of time. c) Determine a function that gives the velocity of the bob as a function of time. d) Determine a function that gives the acceleration of the bob as a function of time. e) Find the max velocity of the bob and the time at which it first occurs.7) A piston in an engine oscillates up and down from a rest position as shown. The motion of this piston can be approximated by the function h (t) = 0.05 cos(13t), where t is rest position time, in seconds, and h is the displacement of the piston head from rest position, in meters, at time t. a) Determine an equation for the velocity of the piston head as a function of time. b) Find the max and min velocities and the times at which they occur. Answers: 1)a) 0.031/min b) 22 min c) N(t) = 100 ()2z d) -2.65 mg/min 2)a)i) 83.3 mg ii) 27.9 mg b)i) -15.2 mg/day ii) -5.1 mg/day c) 7.6 days 3) t = 0s, v = 1m/s 4)a) He will NOT be able to rescue Rocco. b) He must swing back and forth 3.75 times before he can safely drop to the ground. 5) a) max voltage: 170 V at times t, in seconds, t = (, ke Z, k 2 0) min voltage: -170 V at times t, in seconds, t = #+3 , KEZ, k 20 bli) T = -s ii) f = 60 Hz iii) A = 170 V 6)a) 1.42 s b) h(t) = 8 cos(4.43t) c) v(t) = -35.44 sin(4.43t) d) a(t) = -157 cos(4.43t) e) max velocity: 35.4 cm/s at time t = 1.06 s 7) a) v(t) = -0.65 sin(13t) b) max velocity: 0.65 m/s att = {(#+3) , k E Z, k 2 0); min velocity: -0.65 m/s at t = ( 4k+1)I , k EZ, k 20)