A piece of pasture grows at a constant rate every day.?

Question

200 sheep will eat up the grass in 100 days. 150 sheep will eat up the grass in 150 days. How many days does it take for 100 sheep to eat up the grass?

Answer 1

Let x be the number of sheep.

The consumption only depends on the number of sheep, since the growth rate is constant.

Hence dx/dt (consumption rate) = xk

==> dx/x = kdt ==> ∫dx/x = ∫kdt

==> ln x = kt + c1

==> x = ce^(kt) ....... Let e^c1 = c

==> ce^(100k) = 200 ==> c = 200/e^(100k)

==> ce^(150k) = 150 ==> 200/e^(100k).e^(150k) = 150

==> e^(50k) = 3/4  ==> 50k = ln(3/4) ==> k = -.005753641

ce^(-.005753641(100)) = 200

==> c = 200/e^(-0.5753641) = 355.5555

==> x = 355.5555e^(-0.005753641t)

==> 355.5555e^(-0.005753641t) = 100

==> e^(-0.005753641t) = 0.281250043

==> -0.005753641t = ln 0.281250043

==> t = 220.4710326 days