Unit 3 Discussion Assignment The table below provides information taken from a p

Question

Unit 3 Discussion Assignment

The table below provides information taken from a population report by the Atlanta Journal-Constitution (March 20, 2011) based on 2010 U.S. Census results. For each of the five counties in the metro area, it gives an estimate of the population on Jan. 1, 2010, and the percent change between 1/1/2000 and 1/1/2010. If we assume that the percent change for each county is remaining constant, then the population of each county can be modeled using an exponential function with a growth rate given by the annual percentage change.

County                      2010 Population Estimate                 Annual Percent Change (Growth Rate)

Henry

203,922

5.4%   dP/dt = 0.054P

Cherokee

214,346

4.2%     dP/dt = 0.042P

Gwinnett

805,321

3.2%     dP/dt = 0.032P

Cobb

688,078

1.2%     dP/dt = 0.012P

Fulton

920,581

1.2%     dP/dt = 0.012P

Dekalb

691,893

0.38%    dP/dt = 0.0038P

Fayette

106,567

1.6%     dP/dt= 0.016P

Instructions:

First choose one of the counties given in the table and answer the following questions showing all work. Secondly, choose a second county and either post original work * for that county or “double-check” work posted by another student in the class. If you choose to “double check” another answer, you must show work that either corroborates or disputes that answer. For example, an answer such as “I got the same answer” or “I disagree” is not considered substantive and will receive 0 points unless an explanation (with work shown) as to why you agree or disagree is included.

Questions:

Find an exponential function of the form P(t)=P0ekt , which gives the population of the county in the year 2010 + t. (2 points)

Evaluate the function at t = 8 and give a verbal interpretation of your answer. (3 points)

Find the derivative of the function. (2 points)

Evaluate the derivative at t = 8 and give a verbal interpretation of your answer. (3 points)

Henry

203,922

5.4%   dP/dt = 0.054P

Cherokee

214,346

4.2%     dP/dt = 0.042P

Gwinnett

805,321

3.2%     dP/dt = 0.032P

Cobb

688,078

1.2%     dP/dt = 0.012P

Fulton

920,581

1.2%     dP/dt = 0.012P

Dekalb

691,893

0.38%    dP/dt = 0.0038P

Fayette

106,567

1.6%     dP/dt= 0.016P

Explanation / Answer

1) firsrt take country Heny

dp/dt = 0.054*P

dp/p = 0..054*dt

integrating the function we get

ln (p) = 0.054*t + c

taking antilog bothside = P = P0*e 0.054*t ............(a)

at t =0 , p = 203922 putting this in equation ...(a) p0 = 203922 / e = 75018.71

so, population of Henry follow the equation P(t) = 75018.71*e0.054*t

at t =8 , p = 75018.71* e 0.432 = 115553.95 = 115554, poputation has decreased in t=8 year

at t = 8 ,dp/dt = 75018.71*0.054*e0.432 = 6239.913 = 6240 , population has decreased by 6240 persons in 8th year.

2) now take country Cherokee

same as question (1) equation of population will be

P = P0*e0.042*t

at t=0, p = 214346 putting in equation P0 = 78853.49

SO, P = 78853.49*e0.042*t

at t =8 p = 110342 , population is decreased

dp/dt at t =8 , P'(8) = 4634, population is decreaed 4634 person in 8th year

3) coumtry GWINNETT

P = 296261.04*e0.032*t

P(8) = 382696

P'(8) = 12246

4) COUNTRY COBB

P = 253129.75*e0.012*t

P(8) = 278635

P'(8) = 3344

5) FULTON

P = 338662.82*e0.012*t

P(8) = 372786

P'(8) = 4473

6) DEKALB

P = 254533.21*e0.0038*t

P(8) = 262390

P'(8) = 997

7) FAYETTE

P = 39203.8*e0.016*t

P(8) = 44557

P'(8) = 713

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