You have purchased a piece of property with a small lake that is suitable for ra

Question

You have purchased a piece of property with a small lake that is suitable for raising trout. You feel that without any aid from you the trout population introduced into the pond will grow at a nominal annual rate of 5%. You realize the resulting Differential Equation for the growth rate becomes dy/dt = (1/20) y

1) Draw the slope field for this D.E and plot a solution curve through the point that would correspond to the introduction of 50 trout into the pond today, t =0.

2) Find an explicit solution to this D.E.

3) Find the projected undisturbed trout population in the lake after 10 years, 20 years, 30 years, 40 years, and 50 years.

Explanation / Answer

Given dy/dt = kt where k = 1/20

and y'(t) = 5% = 1/20

dy/dt = kt

=> y'(t) = y0kekt

1/20 = y0 (1/20)

y0 = 1

therefor y = e1/20 t

1) y = e1/20 t   + 50

2) y = e1/20 t

3) y(10) = e0.5 = 1.6487

    y(20) = e = 2.7182

    y(30) = 4.4817

    y(40) = 7.389

    y(50) = 12.182

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