m2121t2rev-sol-fs2015 - MATH/2121 CALCULUS FOR THE LIFE...

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MATH/2121, CALCULUS FOR THE LIFE SCIENCES I TEST 2 REVIEW PROBLEMS SOLUTIONS (1) The doubling time for a certain bacteria culture is 12 hr. (a) Determine the exponential growth rate k . k = ln(2) 12 . 058 . (b) If the initial population is 200 what is the population after 10 hours? The population y = y ( t ) at time t hours is given by y = y e kt = 200 e (ln(2) / 12) t bacteria . So the population after 10 hours is y (10) = 200 e (ln(2) / 12)10 = 200 e 5 ln(2) / 6 356 bacteria . (c) How long will it take the population to triple? y = 3 × 200 = 200 e (ln(2) / 12) t = 3 × 200 = e (ln(2) / 12) t = 3 = ln(2) 12 t = ln(3) = t = 12 ln(3) ln(2) 19 hours . (2) Carbon-14 has a half-life of 5,600 years. How old is a piece of wood that has lost 70% of its carbon-14? Letting y = y ( t ) be the amount of carbon-14 after t years we have y = y e kt where y is the initial amount of carbon-14 and k = - ln(2) 5600 . 1
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Our wood has lost 70% of its carbon-14 so 30% remains, that is y = 0 . 3 y at present. Now y e kt = 0 . 3 y = e kt = 0 . 3 = kt = ln(0 . 3) = t = ln(0 . 3) k = - 5600 ln(0 . 3) ln(2) 9727 . That is our piece of wood is about 9,727 years old. (3) Plutonium-24 has a half-life of 13 years. (a) How much of a sample weighing 4 g will remain after 100 years? We have decay constant k = - ln(2) / 13 and the sample weighs y = 4 e - ln(2) t 13 grams after t years. Thus after 100 years our sample weighs y = 4 e - 100 ln(2) 13 0 . 0193 g (b) After what length of time will a 4 g sample have decayed to 3 g? y = 3 = 4 e - ln(2) t 13 = 3 = e - ln(2) t 13 = 3 4 = 0 . 75 = ⇒ - ln(2) t 13 = ln(0 . 75) = t = - 13 ln(0 . 75) ln(2) 5 . 4 years . (4) Convert the angles from degrees to radians: 30 , 45 , 60 , 90 , 180 , - 70 , - 240 , 480 30 = π 6 rad , 45 = π 4 rad , 60 = π 3 rad , 90 = π 2 rad , 180 = π rad , - 70 = - 7 π 18 rad , - 240 = - 4 π 3 rad , 480 = 8 π 3 rad . 2
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(Note: Angles - 240 and 480 are coterminal since 480 = - 240 + 2 × 360.) (5) Convert the angles from radians to degrees: π 6 , π 4 , π 3 , π 2 , π, - 7 π 8 , - 8 π 5 , 23 π 9 π 6 rad = 30 , π 4 rad = 45 , π 3 rad = 60 , π 2 rad = 90 , π rad = 180 - 7 π 8 rad = - 157 . 5 , - 8 π 5 rad = - 288 , 23 π 9 rad = 460 (6) What is the smallest positive angle coterminal with - 20 ? - 9 π/ 8 rad? 340 and 7 π/ 8 rad respectively . (7) Explain how the values sin( θ ) and cos( θ ) are related to a point on the unit circle: We place the angle θ in standard position in the x - y plane. The terminal ray for our angle meets the unit circle x 2 + y 2 = 1 at the point ( x, y ) = (cos( θ ) , sin( θ )).
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