"When zombies finally take over, the population of the Earth will decrease exponentially. Every HOUR that goes by the human population will decrease by 5%. The population today is 6,000,000,000, find a function P that gives the population of the earth d DAYS after the beginning of the zombie takeover."This without a doubt puts me in between a rock and a difficult place. As you may consider I researched this query and numerous methods of exponential boom everywhere on line, in my textbook, and in my class notes, but I nonetheless come complete circle and locate myself lower back to square one, and not using a clue what so ever on how this works. i might really admire it if someone should at least factor me inside the right direction like the way to version a decay with exponential variables regarding HOURS and DAYS because the query says. I already recognize my preliminary method being 6,000,000,000(0.85) I simply need to know the exponents used! Is it 6,000,000,000(0.eighty five)^d/60, 6,000,000,000(zero.eighty five)^d/1? Or some thing like that? Please assist I actually need to understand this idea!

Question

&quot;When zombies finally take over, the population of the Earth will decrease exponentially. Every HOUR that goes by the human population will decrease by 5%. The population today is 6,000,000,000, find a function P that gives the population of the earth d DAYS after the beginning of the zombie takeover.&quot;This without a doubt puts me in between a rock and a difficult place. As you may consider I researched this query and numerous methods of exponential boom everywhere on line, in my textbook, and in my class notes, but I nonetheless come complete circle and locate myself lower back to square one, and not using a clue what so ever on how this works. i might really admire it if someone should at least factor me inside the right direction like the way to version a decay with exponential variables regarding HOURS and DAYS because the query says. I already recognize my preliminary method being 6,000,000,000(0.85) I simply need to know the exponents used! Is it 6,000,000,000(0.eighty five)^d/60, 6,000,000,000(zero.eighty five)^d/1? Or some thing like that? Please assist I actually need to understand this idea!

Monserrat Ellison · Accepted Answer

Since you are doing calculus, we will use the real mathematician&#039;s exponential function       e    x  . Measure time in hours, with   t  =  0 when the exponential decay began. Then                    (1)                    P        (        t        )        =        P        (        0        )                  e                      −            k            t                          ,            (1)where k is a constant. The initial population   6  ×      10    9  .First we find k. We are told that   P  (  1  )  =  (  0.95  )  P  (  0  ). Substituting in (1), we obtain  (  0.95  )  P  (  0  )  =  P  (  0  )      e          −      k        .Do some cancellation, and take the natural logarithm of both sides. We obtain   k  =  −  ln  ⁡  (  0.95  ).Now Equation (1) can be rewritten as                    (2)                    P        (        t        )        =        P        (        0        )                  e                      −            (            −            t            ln            ⁡            (            0.95            )            )                          =                  e                      t            ln            ⁡            (            0.95            )                          =        P        (        0        )        (        0.95                  )          t                .             (2)Note that we could have obtained the form   P  (  t  )  =  P  (  0  )  (  0.95      )    t   far faster. For every hour that goes by, the population gets multiplied by   0.95.When d days have elapsed,   24  d hours have elapsed. Set   t  =  24  d in either version of (2).

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