The population of a community is known to increase at a rate proportional to the

Question

The population of a community is known to increase at a rate proportional to the number of people present at time t. The initial population P0 has doubled in 5 years.

1)Suppose it is known that the population is 9,000 after 3 years. What was the initial population P0? (Round your answer to one decimal place.) P0 =

2)What will be the population in 10 years? (Round your answer to the nearest person.) persons.

3)How fast is the population growing at t = 10? (Round your answer to the nearest person.) persons/year

I am not sure how to approach this problem, I know it's proportional so I can use the formula P=Poe^kt ? Could you explain your steps please?

Thank you

Explanation / Answer

dP/dt = P(t)
P = ? P(t) dt
P = e^(kt)

So

P(t) = P0*e^(kt)

If P(5) = 2P0:

P(5) = P0*e^(k5)
2P0 = P0*e^(5k)
2 = e^(5k)
ln(2) = 5k
k = ln(2)/5

P(t) = P0*e^(ln(2)t/5)
P(t) = P0*2^(t/5)

2) P(t) = 9,000 when t = 3

P(3) = 9,000
P0*2^(3/5) = 9,000
P0 = 9,000/(2^(3/5))
P0 = 5937.8

P(10) = 5937.8 *2^(10/5)
P(10) = 5937.8 *2

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