N(t)=200e^(kt)

N(3) = 500

200e^(k*3) = 500

First we will find k by solving:

200e^(k*3) = 500 divide both sides by 200

e^(3k) = 2.5

(e^k)^3 = 2.5

Let t represent the time in hours needed for the original population to triple in size:

N(t) = 3N(0)

200e^(kt) = 3*200e^(k*0)

200e^(kt) = 600 divide both sides by 200

(e^k)^t = 3

(e^k)^(3t) = 3^3

((e^k)^3)^t = 27

2.5^t = 27

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t = 3.6 hr

The time required for the original population to triple in size is 3.6 hours.