Let w be the number of worms (in millions) and r the number of robins (in thousa
ID: 2878568 • Letter: L
Question
Let w be the number of worms (in millions) and r the number of robins (in thousands) living on an island. Suppose w and r satisfy the following differential equations, which correspond to the slope field shown below. dw/dt=w - wr, dr/dt = -r + wr. Suppose that initially there are no robins and 2.5 (million) worms on the island. Solve the differential equations in this case w(t) = Now suppose that initially there are no worms and 1 (thousand) robins on the island. Solve the differential equations in this case. w(t) = r(t) =Explanation / Answer
Solution:
If initially there are no robins, there will never be any robins. There is absolutely no point in looking at the robin equation;
w(0) = initial number of worms = 2.5 million ; r(0) = 0
dw/dt = w since r(t) = 0.
w(t) = w(0) e^t = 2.5 e^t
r(t) = 0 then satisfies the second equation with the initial condition r(0) = 0.
2nd system:
No worms: w(t) = 0. Then the robins have nothing to eat and starve to death:
dr/dt = - r ,
r(t) = r(0) e^(-t) = 1000 e^(-t)
Eventually this is < 1, which means no more robins.
Specifically r(0) e^(-t) < 1 means ln(r(0)) - t < 0 or t > ln r(0) = ln1000 = 6.91 units
The unit of time is not specified. If you just look at r(0) e^(-t) , this is positive but decreases rapidly as t --> infinity.
So
w(t) = 0 and r(t) = 1000 e^(-t)
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