10.27.Solutions by hand WS 3.3 - MA 160 Section 3.3...

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Unformatted text preview: MA 160 Section 3.3 Worksheet Name 6:. G as; t'am A - - iications: Uninhibited and Limited Growth Models. E ntial Gr wth. “any ,m y=Ce'“ W kt ° . . dr P(t) = P0 e where k is the exponential growth rate. In 1 and 2, find the eneral form of the fimction that satisfies the iven difi'erential - uation: fl d1- 3. Approximately 10,000 bacteria are placed in a culture. Let P(t) be the number of bacteria present in the culture after t hours, and suppose that P(t) satisfies the differential equation P’U) = 0. 55P(t) . :1. Find the formula for P(t). b. What is P(0)? F1473: {game-- e asst .5519) P(0): io’me — t d. What is the exponential growth rate ? K zeta“; c. How many bacteria are there after 5 hours? 49(5) : to we 6";56). ’ - fféj‘faé, f. What is the size of the bacteria culture when it is growing at the rate of 34,000 bacteria per hour? P’Cf): 052; Purl 34am : ~56" PC?) 34,6199 .5; e. Use the differential equation to determine how fast the bacteria culture is growing when it reaches 100,000. P‘Zé) :3 a $500,009) 3" 5-5-5342 We. Pelt/«gm 4. Suppose a population is growing with exponential growth rate 0.04. In how many years will the current population triple? Peg-1 :, Pa 6 If t’ t E 3 3P0 : Paeoa‘tt 3 to‘tt 1" "a C «911.3: wattt': M3 '0‘} ; é 5. How much money must you invest now at 6% interest compounded continuously in order to have $5000 at the end of 3 cars? - , K e- Y P (:t ) a Pa (2 n- n.- .09 ; .0G(3\ 5000::- Poe 30.; 5m ‘- Fa «06(3) 500a $5 “W:Pa 0L Po 2 ‘H’ve.35‘ e gaawm £4€I7éo3§Wa~¥vmdéc )fi'gjzaa-e 3500:: l W3gwwa «fWaeh CW6 W 3 a 3 6. The population of the United States in the year 1776 was about 2.5 million. In its bicentennial year, 1976, the population was 216 million. Assume that at any time, the population grows at a rate that is proportional to the I o ulation at that time. a. Find the exponential growth rate and round your b. Assuming 1? 76 corresponds to t = 0, what is the answer to 4 decimal places. t =- 0 M i"! 7 a, function that models this situation? c. What is the estimated U.S. population for 2010? t -.-.-. .2010 [email protected] :2 23‘! P523“: n.5'e‘09‘303“ d. Using the differential equation corresponding 6- In what War Will this 1901911151’6101‘:l “33011 350 million? to this problem, what is the growth rate in 2010? 3‘50 5‘ at 5 e ' 0 lag t P’Hzlzeogast Pa) 350 gonna ,s east 7. $5000 is deposited into a savings account at 3% compounded continuously. a. Wl/i‘at is the formula for A(t), the balance after t years? b. What differential equations is satisfied by A(t) ? (f):[email protected] 8803f: A’Cf):o03A(éJ c. How much will be in the account alter 4 years? d. When will the balance reach $8,000? A(q) : Socoe.03(‘ll: gas)?! (1,? :_Hb;i:3i.03f 5-000 east g: = 8 fi=i5',(o’7 e. flowfastis the balance growing when it reaches $8,000? Jone?) =lm~3+ game. A (8:..03‘Eidél]: .03(§coa):$¢)qfl.m Mtg!) W4.— ’ 9 g :s f 8. An amount of money is deposited in a savings account with interest compounded continuously. Let A(t) be the balance in the accOunt after t years. Match each of the following questions with its corresponding solution. Solution A. A(t) = Pe ’ ‘ B. A(0) C. Find MS) D. Find A'(5) E. Solve A(t) = 5 for t . F. Solve A'(i)=5 fort. D a. How faml the balance be growing in 5 years? A b. Give the general form of the function A(t) H o. How long will it take the initial deposit to grow to five times the initial deposit? C (I. Find the balance after 5 years. The. a?» 9 C 53 E e. When will the balance be $5? F_ f. When will the balance be growing at the rate of $5 per year? 5 g. What was the principal amount? L" W '—' 5 Q h. Give a differential equation satisfied by A(t) G. y' = ry or A’(i) = rAU) H. Solve A(t) = 5*A(0) fort . ...
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