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InterviewSolution
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If the population of country double in 50 years, in how many years willit triple under the assumption that the rate of increase is proportional tothe number of inhabitants. |
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Answer» Let x denote the population at time t in years. Then `(dx)/(dt) propto x rArr (dx)/(dt) = kx`, Where k is constant of proportionality. Solving `(dx)/(dt) = kx`, we get `int(dx)/x= intkdt` or `logx=kt + c` or `x=e^(kt+c)` or `x=x_(0)e^(kt)`, Where `x_(0)` is the population at time t=0. Since it doubles in 50 years, at t=50, we must have `x=2x_(0)` Hence, `2x_(0)=x_(0)e^(50k)` or `50k = log2` or `k=(log2)/(50)` so that `50k =log2` or `k=(log2)/(50)` so that `x=x_(0)e^((log2)/50)t` or `t=(50 log3)/(log 2)=50 log_(2)(3)` |
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