3. -1 points SApCalcBr1 6.4.025. Suppose B is a function of time t and B always

Question

3. -1 points SApCalcBr1 6.4.025. Suppose B is a function of time t and B always increases at a rate inversely proportional to its value. Write a differential equation that describes this relationship. (Let k represent the constant of proportionality.) dB dt Submit Answer Save Progress 4.12 points SApCalcBr1 6.4.027 Suppose a population always grows at a rate equal to 1/200 of the population size (measured in members per year). Write a differential equation that represents this relationship. (Let P(t) represent the population t years from now.) dP dt If the population is currently 80,000, find a solution to the differential equation that gives the population t years from now. P(t)

Explanation / Answer

dB/dt proportional to 1/B

So, dB/dt = k/B ---> ANS

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dP/dt = 1/200 * P ---> ANS

Variable separable :
dP/P = dt/200

Integrating :
ln|P| = t/200 + C

When t = 0 , P = 80000 :
ln(80000) = 0/200 + C

C = ln(80000)

So, we have
ln|P| = t/200 + ln(80000)

ln(P) - ln(80000) = t/200

ln(P/80000) = t/200

P/80000 = e^(t/200)

So,
P = 80000e^(t/200) ----> ANS

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