The population of a certain city was 112000 in 2016 and the

The population of a certain city was 112,000 in 2016, and the observed doubling time for the population is 25 years. a) Find an exponential model n(t) n_0 2^18/a for the population t years after 2016 b) Find an exponential model n(t) = n_oe^n for the population t years after 2016 c) Estimate when the population will reach 300,000 The half-life of radium-226 is 1600 years. Suppose we have a 35-mg sample. a) find a function m(t) m_0e^rt that models the mass remaining after t years. b) How much of the sample will remain after 3000 years? c) After how long will only 10 mg of the sample remain? Find the reference number (t-bar) for each value of t and the terminal point (x,y) on the units determined by t. a) t = - 4 pi/3 b) t = 17 pi/6

Solution

1)

a) in 2016, t =0

at t=0, n(0)=112000

at t =25,n(25)=2*112000

2*112000=112000*2^25/a

2^25/a=2

25_/a=1

a=25

n(t)=112000*2^t/25

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b)n(t)=112000*2^t/25

n(t)=112000*(2^1/25)^t

n(t)=112000*(e^ln21/25)^t

n(t)=112000*(e^(1/25)ln2)^t

n(t)=112000*(e^0.027725887)^t

c) population reaches 300000

112000*(e^0.027725887)^t=300000

(e^0.027725887)^t=300/112

0.027725887t= ln(300/112)

t =(1_/0.027725887)*ln(300/112)

t =35.5366

year =2016+35.5366 =2051.5366

population reaches 300,000 in year 2052