The number of Calories per day that a person must consume to maintain his weight

Question

The number of Calories per day that a person must consume to maintain his weight is proportional to his current weight. The rate at which the person's weight changes is proportional to the difference between the number of Calories x that he consumes per day and the number of Calories required to maintain his weight. Write a differential equation for the weight w of the person.

Explanation / Answer

Cal cons per day = C*dt Cal excess = (C - 20*W )*dt Weight change = [(C - 20*W ) / 3500]*dt = dW dW/dt = (C - 20*W ) / 3500 this is the differential equation. To solve, separate variables dW/[(C - 20*W )] = dt/3500 Integrate -1/20 * ln(C - 20*W) = t/3500 + K ln(C - 20*W) = -(20/3500)*t - K C - 20*W = K'*e^-(20/3500)*t 20*W = C - K'*e^-(20/3500)*t W = C/20 - K''*e^-(20/3500)*t If the initial weight is 160, then W = 160 at t = 0: 160 = C/20 - K''; K'' = C/20 - 160 W = C/20 -(C/20 - 160) *e^-(20/3500)*t If C/20 > 160, the coefficient of the exponential is negative and weight decreases. 3000/20 = 150. so weight will increase. For a steady-state condition, C/20 - 160 must be 0.

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