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the initial termite population of a house is 1000 and is growing exponentially at a yearly rate of 30%. how long will it take for the termite population to double?

Let the termite population be X.

Then X =Xo * e^(kt), where Xo=1000 (initial population) and t denotes the time in years.

= 1000 e^(kt)

To discover the value of k, when t=1, X= 1000* 1.3 =1300

Hence, 1300 = 1000 e^(k)

k =ln (1300/1000) =0.262

ie X =1000 e^(0.262 t)

When the population doubles, X=2000

2000= 1000 e^(0.262 t)

2= e^(0.262 t)

ln2 = 0.262 t

Hence t= ln2 / 0.262 =2.642 years (shown)

[In fact, if you studied nuclear physics, you will realize the above is merely related to the half life concept, where half life is equals to ln2 /k ]

Hope this helps. Peace.

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