Advanced MathThe Gompertz model has been used to model population growth.
4 weeks ago
The Gompertz model has been used to model population growth. (dy)/(dt) = ry ln((K)/(y)), where r= 0.73 per year, k= 33,800 kg, (y_0)/(k) = 0.27 Use the Gompertz model to find the predicted value of y(3) Round your answer to the nearest integer
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halimanasher12 Staff answered 4 weeks ago
Consider the following differential equation: (dy)/(dt) = ry ln ((K)/(y)) where r= 0.73 per year, k= 33,800 kg, (y_0)/(k) = 0.27 The objective is to find the predicted value of y(3) by using Gompertz model. Given (dy)/(dt) = ry ln ((K)/(y)), it is variable separable equation as r and K are constant. This implies, (dy)/(y ln (K)/(y)) = rdt Integrate on both sides, int (dy)/(y ln (K)/(y)) = int rdt int (dy)/(y ln (K)/(y)) = rt + c where c is an arbitrary constant. Now, use u-substitution method as follows: u = (K)/(y) du = = (-K)/(y^2) dy dy = ((-y^2)/(K))du This implies, int (1)/(y ln (u)) ((-y^2)/(K))du = rt + c int = (-y)/(K ln(u)) du = rt + c int (-((K)/(u)))/(K ln(u))du = rt + c int (-1)/(u ln (u))du = rt + c Assume v = ln (u) which implies, dv = (1)/(u)du du = udv Substitute this in the above integration to get, int (-1)/(v)dv = rt + c -ln(v) = rt + c -ln(ln(u)) = rt + c (v = ln(u)) ln(u)=ce^(-rt) u = e^(ce^(-rt)) (K)/(y) = e^(ce^(-rt)) y = (K)/(e^(ce^(-rt))) Since r = 0.73, k = 33.800, (y_0)/(k) = 0.27 This implies y_0 = 9126 at t= 0, substitute the given value in the expression y=(K)/(e^(ce^(-rt))). 9126 = (33800)/(e^(ce^(0.73(0)))) 9126 = (33800)/(e^(ce^(0))) e^c = (33800)/(9126) c = ln((100)/(27)) c=1.309 Now, we find the value of y(3), substitute all the known values in the expression y = (K)/(e^(ce^(-rt))). y = (33800)/(e^(1.309)e^((-0.73)(3))) =29193.96 ~ 29194 Therefore, the value of y(3) by using Gompertz model is 29194.
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