in 2000 the population of country A reached 25 million and i

in 2000 the population of country A reached 2.5 million, and in 2025 it is projected to be 4.5 million. (a) Find values for Po and a so that the following formula models the population of country A in year x f(x)=Poax-2000 (b) Estimate the country\'s population in 2010 to the nearest hundredth of a million (c) Use f to determine the year during which the country\'s population might reach 7 million. (a) Find values for Po and a Po=million Round to the nearest hundredth as needed.) a= (Round to five decimal places as needed.) (b) The population in 2010 will be million. (Use the rounded answer from part (a) to find this answer. Round to the nearest hundredth as needed.) (c) The country\'s population might reach 7 million around (Use the rounded answer from part(a) to find this answer. Round to the nearest year.)

Solution

(a)

f(x)=P_oa^x-2000

f(2000) =2.5 million

=>P_oa^2000-2000=2.5 million

=>P_oa⁰=2.5 million

=>P_o*1=2.5 million

=>P_o=2.5 million

f(x)=2.5a^x-2000 million

f(2025)=4.5 million

=>2.5a^2025-2000 =4.5

=>2.5a²⁵ =4.5

=>a²⁵ =4.5_/2.5

=>a=(4.5_/2.5)^1/25

=>a=1.02379

f(x)=2.5*1.02379^x-2000 million

(b)

population in 2010 = f(2010)

population in 2010 = 2.5*1.02379^2010-2000

population in 2010 = 2.5*1.02379¹⁰

population in 2010 = 3.16

the population in 2010 will be 3.16 million

(c)

when population reaches 7 million

f(x)=7

=>2.5*1.02379^x-2000 =7

=>1.02379^x-2000 =(7/2.5)

=>x-2000 =ln(7/2.5) _/ln(1.02379)

=>x=2000+(ln(7/2.5) _/ln(1.02379))

=>x=2043.7922971241315177138603359527

the country\'s population might reach 7 million around 2044

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