The population of a certain is known to increase at a rate proportional to the number of people presently living in the country. If after two years the population has doubled, and after three years the population is 20,000 estimates the number of people initially living in the country.

The population of a certain is known to increase at a rate proportiona

1.	The population of a certain is known to increase at a rate proportional to the number of people presently living in the country. If after two years the population has doubled, and after three years the population is 20,000 estimates the number of people initially living in the country.
Answer» Let N denote the number of people living in the country at any time t, and let `N_(0)` denote the number of people initially living in the country. Then, from `(dN)/(dt) propto N, (dN)/(dt)-kN=0` which has the solution, `N=ce^(kt)`…………………..(1) At `t=0, N=N_(0)`, hence, equation (1) states that `N_(0)=ce^(k(0))`, or `c=N_(0)`. Thus, `N=N_(0)e^(kt)`....................(2) At `t=2, N=2N_(0)`. Substituting these values into equation (2), we have `2N_(0)=N_(0)e^(2k)`or `e^(k)=sqrt(2)`. Substituting this value into equation (1) gives `N=N_(0)(sqrt(2))^(k)` ...............(3) At `t=3, N=20,000`. Substituting these values into equation (3), we obtain `N_(0)=20,000//2sqrt(2)=7071`

Discussion

No Comment Found

Related InterviewSolutions

Find the general solution of the differential equations:`(1+x^2)dy+2x y dx=cotx dx(x!=0)`
If `y=x/(In|cx|)` (where c is an arbitrary constant) is the general solution of the differential equation `(dy)/(dx)=y/x+phi(x/y)` then function `phi(x/y)` is:A. `x^(2)//y^(2)`B. `-x^(2)//y^(2)`C. `y^(2)//x^(2)`D. `-y^(2)//x^(2)`
`y(1+logx)(dx)/(dy)-xlogx=0, y=e^(2)" if "x=w`A. `y=exlogx`B. `ey=xlogx`C. `x=eylogy`D. `ex=ylogy`
Find the general solution of the differential equations:`xlogx(dx)/(dy)+y=2/xlogx`
`e^(dy//dx)=1`A)`y=x+c`B)`y=e^(x)+c`C)`y=c`D)`y=cx+d`A. `y=x+c`B. `y=e^(x)+c`C. `y=c`D. `y=cx+d`
`(dy)/(dx)=(y)/(x)-x/(y),` where `y=vx` A)`y^(2)=xlog((c)/(x))` B)`y^(2)=4xlog((c)/(x))` C)`y^(2)=2x^(2)log((c)/(x))` D)`v^(2)=2clogv`A. `y^(2)=xlog((c)/(x))`B. `y^(2)=4xlog((c)/(x))`C. `y^(2)=2x^(2)log((c)/(x))`D. `v^(2)=2clogv`
`e^(dy//dx)=x`A)`y=xlog((x)/(e))+c`B)`y=xlogx+c`C)`y=e^(x)+c`D)`x=ylog(ex)+c`A. `y=xlog((x)/(e))+c`B. `y=xlogx+c`C. `y=e^(x)+c`D. `x=ylog(ex)+c`
The differential equations, find a particular solution satisfying the given condition: `(x^3+x^2+x+1)(dy)/(dx)=2x^2+x ; y=1`when `x = 0`
Find the general solution of the differential equations `(dy)/(dx)=sin^(-1)x`
The differential equation of the family of curves `y=e^x(Acosx+Bsinx),`where `A`and `B`arearbitrary constants is(a) `( b ) (c) (d)(( e ) (f) d^(( g )2( h ))( i ) y)/( j )(( k ) d (l) x^(( m )2( n ))( o ))( p ) (q)-2( r )(( s ) dy)/( t )(( u ) dx)( v ) (w)+2y=0( x )`(y)(z) `( a a ) (bb) (cc)(( d d ) (ee) d^(( f f )2( g g ))( h h ) y)/( i i )(( j j ) d (kk) x^(( l l )2( m m ))( n n ))( o o ) (pp)+2( q q )(( r r ) dy)/( s s )(( t t ) dx)( u u ) (vv)+2y=0( w w )`(xx)(yy)`( z z ) (aaa) (bbb)(( c c c ) (ddd) d^(( e e e )2( f f f ))( g g g ) y)/( h h h )(( i i i ) d (jjj) x^(( k k k )2( l l l ))( m m m ))( n n n ) (ooo)+( p p p ) (qqq)(( r r r ) (sss) (ttt)(( u u u ) dy)/( v v v )(( w w w ) dx)( x x x ) (yyy) (zzz))^(( a a a a )2( b b b b ))( c c c c )+y=0( d d d d )`(eeee)(ffff)`( g g g g ) (hhhh) (iiii)(( j j j j ) (kkkk) d^(( l l l l )2( m m m m ))( n n n n ) y)/( o o o o )(( p p p p ) d (qqqq) x^(( r r r r )2( s s s s ))( t t t t ))( u u u u ) (vvvv)-7( w w w w )(( x x x x ) dy)/( y y y y )(( z z z z ) dx)( a a a a a ) (bbbbb)+2y=0( c c c c c )`(ddddd)A. `(d^(2)y)/(dx^(2))-2(dy)/(dx)+2y=0`B. `(d^(2)y)/(dx^(2))+2(dy)/(dx)-2y=0`C. `(d^(2)y)/(dx^(2))+((dy)/(dx))^(2)+y=0`D. `(d^(2)y)/(dx^(2))-7(dy)/(dx)+2y=0`

The population of a certain is known to increase at a rate proportional to the number of people presently living in the country. If after two years the population has doubled, and after three years the population is 20,000 estimates the number of people initially living in the country.

Discussion

No Comment Found

Related InterviewSolutions

Reply to Comment