This problem is causing me too much stress, any help would beappreciated: Let P(

Question

This problem is causing me too much stress, any help would beappreciated: Let P(t) be the population of a country, in millions, t yearsafter 2000. In year 7 the population was 300.2 million. In 2009 thepopulation was 303.9 million. a) Find a formula for P(t) assuming it is linear. b) Find a formula for P(t) assuming it is exponential. ( myguess was P(t) = 300.2(1.012325117)^t) c) Estimate the year in which both models will surpass 310million people. Answer must be algebraic. This problem is causing me too much stress, any help would beappreciated: Let P(t) be the population of a country, in millions, t yearsafter 2000. In year 7 the population was 300.2 million. In 2009 thepopulation was 303.9 million. a) Find a formula for P(t) assuming it is linear. b) Find a formula for P(t) assuming it is exponential. ( myguess was P(t) = 300.2(1.012325117)^t) c) Estimate the year in which both models will surpass 310million people. Answer must be algebraic.

Explanation / Answer

( a ) ( 7, 300.2 ) ( 9, 303.9 ) m = ( 303.9 - 300.2 ) / ( 9 - 7 ) = 3.7 / 2 = 1.85 y = mx + b = 1.85x + b y( 7 ) = 300.2 => 1.85( 7 ) + b = 300.2 => 12.95 + b = 300.2 => b = 300.2 - 12.95 = 287.25 => y = 1.85x + 287.25 y( x ) = 310 => 1.85x + 287.25 = 310 => 1.85x = 310 - 287.25 = 22.75 => x = 22.75 / 1.85 = 12.2972973 Final answer: The population surpasses 310 million in the thirteenth year. ( b ) ( 7, 300.2 ) ( 9, 303.9 ) y( x ) = Cekx Ce9k = 303.9 Ce7k = 300.2 Divide the first equation by the second equation. e2k = 303.9 / 300.2 = 1.01232512 => ln e2k = ln 1.01232512 => 2k ln e = ln 1.01232512 => 2k = ln 1.01232512 => k = ( 1 / 2 ) ln 1.01232512 => k = ln sqrt( 1.01232512) => k = ln 1.00614369 => y( x ) = Ce( ln1.00614369)x => y( x ) = C( eln1.00614369)x => y( x ) = C( 1.00614369 )x y( 7 ) = 300.2 => C( 1.00614369 )7 = 300.2 => C( 1.04380664 ) = 300.2 => C = 300.2 / 1.04380664 = 287.60116 => y( x ) = 287.60116( 1.00614369 )x y( x ) = 310 => 287.60116( 1.00614369 )x = 310 => ( 1.00614369 )x = 310 / 287.60116 = 1.07788161 => ln ( 1.00614369 )x = ln 1.07788161 x ln 1.00614369 = ln 1.07788161 => x = ln 1.07788161 / ln 1.00614369 = 12.244724 Final answer: The population surpasses 310 million in the thirteenth year.

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