In order to have population increasing, a country accepts A people a year from other counntries, and the number x(t) of people in a country changes by equation x ′ ( t ) = − 0.03 x ( t ) + Awhere t is time in years. while x ( 0 ) = 16 m and x ( − 10 ) = 15 m find A.This is what I did:from the equation I know that each year the population of the country decrease in 0.03I tried to use the data, since t is in years x ( t 0 ) ∗ 1.03 = x ( t 0 + 1 )Since we wish in 10 years the total popluation increase in million people therefore : x ( t 0 ) ∗ 1.03 = x ( t 0 + 1 ) + 100 , 000I get the feeling my solution is wrong, But I don't really find other ways to approach this.Any ideas? I'll be glad if someone could tell me why my solution fails.

Question

In order to have population increasing, a country accepts A people a year from other counntries, and the number x(t) of people in a country changes by equation      x    ′    (  t  )  =  −  0.03  x  (  t  )  +  Awhere t is time in years. while   x  (  0  )  =  16  m and   x  (  −  10  )  =  15  m find A.This is what I did:from the equation I know that each year the population of the country decrease in 0.03I tried to use the data, since t is in years  x  (      t    0    )  ∗  1.03  =  x  (      t    0    +  1  )Since we wish in 10 years the total popluation increase in million people therefore :  x  (      t    0    )  ∗  1.03  =  x  (      t    0    +  1  )  +  100  ,  000I get the feeling my solution is wrong, But I don&#039;t really find other ways to approach this.Any ideas? I&#039;ll be glad if someone could tell me why my solution fails.

Antwan Wiley · Accepted Answer

Step 1The provided information is                    (1)                              x          ′                (        t        )        =        −        0.03        x        (        t        )        +        A                                (2)                    x        (        0        )        =        16        m                         and                         x        (        −        10        )        =        15        m            Your attempted solution implies that the value of x(t) only changes once per year. If so, I would assume something like a linear difference equation would have been provided. However, since (1) involves a derivative, it seems x(t) is meant to be considered a continuous function instead. If this assumption is correct, then note that (1) shows an exponential decay along with (I assume) a fixed increase value. Note the equation implies using something where the derivative is a multiple of itself, which the exponential function satisfies. A general solution would be of the form                    (3)                    x        (        t        )        =        a                  e                      b            t                          +        c            for real constants a,b and c. Substituting (3) into (1) gives                    (3)                    x        (        t        )        =        a                  e                      b            t                          +        c            Step 2Since this must hold for all t, this means the coefficients of the       e          b      t       and constant terms must be the same on both sides of the equation, i.e.,                    (5)                    a        b        =        −        0.03        a                        ⇒                        a        =        0                 or                 b        =        −        0.03                                (6)                    0        =        −        0.03        c        +        A                        ⇒                        c        =                  A          0.03                    Note that if   a  =  0, then (3) shows that x(t) is a constant function, but (2) shows it&#039;s not, so   b  =  −  0.03 must be the case instead in (5). Thus, (3) becomes                    (7)                    x        (        t        )        =        a                  e                      −            0.03            t                          +                  A          0.03                    Using the provided values, you can now substitute   t  =  0 and   t  =  −  10 into (7) to get 2 equations in the 2 unknowns of a and A. You can then solve these equations to get A. I trust you can do these remaining calculations yourself.

A country accepts. A people a year from other countries, we know that: x′(t)=-0.03x(t)+A while x(0)=16m and x(-10)=15m. Find A

Answered question

Answer & Explanation

New Questions in College Statistics